%I A001700 M2848 N1144
%S A001700 1,3,10,35,126,462,1716,6435,24310,92378,352716,1352078,5200300,
%T A001700 20058300,77558760,300540195,1166803110,4537567650,17672631900,
%U A001700 68923264410,269128937220,1052049481860,4116715363800,16123801841550
%N A001700 C(2n+1, n+1): number of ways to put n+1 indistinguishable balls into
n+1 distinguishable boxes = number of (n+1)-st degree monomials in
n+1 variables = number of monotone maps from 1..n+1 to 1..n+1.
%C A001700 To show for example that C(2n+1, n+1) is the number of monotone maps
from 1..n+1 to 1..n+1, notice that we can describe such a map by
a nondecreasing sequence of length n+1 with entries from 1 to n+1.
The number k of increases in this sequence is anywhere from 0 to
n. We can specify these increases by throwing k balls into n+1 boxes,
so the total is Sum_{k=0..n} C((n+1)+k-1, k) = C(2n+1, n+1).
%C A001700 Also number of ordered partitions (or compositions) of n+1 into n+1 parts.
E.g. a(2)=10: 003 030 300 012 021 102 120 210 201 111 - Mambetov
Bektur (bektur1987(AT)mail.ru), Apr 17 2003
%C A001700 Also number of walks of length n on square lattice, starting at origin,
staying in first and second quadrants - David W. Wilson (davidwwilson(AT)comcast.net),
May 05, 2001. E.g. for n = 2 there are 10 walks, all starting at
0,0: 0,1->0,0; 0,1->1,1; 0,1->0,2; 1,0->0,0; 1,0->1,1; 1,0->2,0;
1,0->1,-1; -1,0->0,0; -1,0->-1,1; -1,0->-2,0.
%C A001700 Also total number of leaves in all ordered trees with n+1 edges.
%C A001700 Also number of digitally balanced numbers [A031443] from 2^(2n+1) to
2^(2n+2). - Naohiro Nomoto (6284968128(AT)geocities.co.jp), Apr 07
2001
%C A001700 Also number of ordered trees with 2n+2 edges having root of even degree
and nonroot nodes of outdegree 0 or 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Aug 02 2002
%C A001700 Also number of paths of length 2*d(G) connecting two neighboring nodes
in optimal chordal graph of degree 4, G(2*d(G)^2+2*d(G)+1, 2d(G)+1),
where d(G) = diameter of graph G. - S. Bujnowski (slawb(AT)atr.bydgoszcz.pl),
Feb 11 2002
%C A001700 Define an array by m(1,j)=1, m(i,1)=i, m(i,j)=m(i,j-1)+m(i-1,j); then
a(n)=m(n,n) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 07 2002
%C A001700 Also the numerator of the constant term in the expansion of Cos^2n(x)
or Sin^2n(x) when the denominator is 2^(2n-1). - rgwv
%C A001700 Consider the expansion of cos^n(x) as a linear combination of cosines
of multiple angles. If n is odd, then the expansion is a combination
of a*cos((2k-1)*x)/2^(n-1) for all 2k-1<=n. If n is even, then the
expansion is a combination of a*cos(2k*x)/2^(n-1) terms plus a constant.
"The constant term, [a(n)/2^(2n-1)], is due to the fact that [cos^2n(x)]
is never negative, i.e. electrical engineers would say the average
or 'dc value' of [cos^2n(x)] is [a(n)/2^(2n-1)]. The dc value of
[cos^(2n-1)(x)] on the other hand, is zero because it is symmetrical
about the horizontal axis, i.e. it is negative and positive equally."
Nahin[62] - rgwv Aug 01 2002
%C A001700 Also number of times a fixed Dyck word of length 2k occurs in all Dyck
words of length 2n+2k. Example: if the fixed Dyck word is xyxy (k=2),
then it occurs a(1)=3 times in the 5 Dyck words of length 6 (n=1):
(xy[xy)xy], xyxxyy, xxyyxy, x(xyxy)y, xxxyyy (placed between parentheses).
- Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 02 2003
%C A001700 G.f. is C(x)/sqrt(1-4x) where C(x) is g.f. for Catalan numbers A000108.
%C A001700 a(n+1) is the determinant of the n X n matrix m(i,j)=binomial(2n-i,j)
- Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 26 2003
%C A001700 a(n-1) = (2n)!/(2*n!*n!), formula in [Davenport] used by Gauss for the
special case prime p = 4*n+1: x = a(n-1) mod p and y = x*(2n)! mod
p are solutions of p = x^2 + y^2. - Frank Ellermann. Example: For
prime 29 = 4*7+1 use a(7-1) = 1716 = (2*7)!/(2*7!*7!), 5 = 1716 mod
29 and 2 = 5*(2*7)! mod 29, then 29 = 5*5 + 2*2.
%C A001700 a(n)=sum{k=0..n+1, binomial(2n+2,k)*cos((n-k+1)*pi)} - Paul Barry (pbarry(AT)wit.ie),
Nov 02 2004
%C A001700 The number of compositions of 2n, say c_1+c_2+...c_k=2n, satisfy that
Sum_(i=1..j)c_i <2j for all j=1..k, or equivalently, the number of
subsets, say S, of [2n-1]={1,2,...2n-1} with at least n elements
such that if 2k is in S, then there must be at least k elements in
S smaller than 2k. E.g. a(2)=3 because we can write 4=1+1+1+1=1+1+2=1+2+1
- Ricky X. F. Chen (ricky_chen(AT)mail.nankai.edu.cn), Jul 30 2006
%C A001700 a(n) = A122366(n,n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Aug 30 2006
%C A001700 a(n) = A000984+A001791. Example: 1+0=1 2+1=3 6+4=10 20+15=35 70+56=126
etc... - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 23 2007
%C A001700 The number of walks of length 2n+1 on an infinite linear lattice that
begin at the origin and end at node (1). Also the number of paths
on a square lattice from the origin to (n+1,n) that use steps (1,
0) and (0,1). Also number of binary numbers of length 2n+1 with n+1
1's and n 0's. - Stefan Hollos (stefan(AT)exstrom.com), Dec 10 2007
%C A001700 If Y is a 3-subset of an 2n-set X then, for n>=3, a(n-1) is the number
of n-subsets of X having at least two elements in common with Y.
- Milan R. Janjic (agnus(AT)blic.net), Dec 16 2007
%C A001700 Also the number of rankings (preferential arrangements) of n unlabeled
elements onto n levels when empty levels are allowed. - Thomas Wieder
(thomas.wieder(AT)t-online.de), May 24 2008
%C A001700 Also the Catalan transform of A000225 shifted one index, ie, dropping
A000225(0). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov
11 2008]
%C A001700 With offset 1. The number of solutions in nonnegative integers to X1+X2+...+Xn=n.
The number of terms in the expansion of (X1+X2+...+Xn)^n. The coefficient
of x^n in the expansion of (1+x+x^2+...)^n. The number of distinct
image sets of all functions taking [n]into[n]. [From Geoffrey Critzer
(critzer.geoffrey(AT)usd443.org), Feb 22 2009]
%C A001700 The Hankel transform of the aerated sequence 1,0,3,0,10,0,... is 1,3,
3,5,5,7,7,.... (A109613(n+1)). [From Paul Barry (pbarry(AT)wit.ie),
Apr 21 2009]
%C A001700 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009:
(Start)
%C A001700 Equals INVERT transform of the Catalan numbers starting with offset 1.
%C A001700 E.g.: a(3) = 35 = (1, 2, 5) dot (10, 3, 1) + 14 = 21 + 14 = 35. (End)
%D A001700 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001700 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001700 M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008),
2544-2563.
%D A001700 E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes
in a rectangle, Discr. Math., 298 (2005). 62-78.
%D A001700 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal
Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A001700 A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining
convex permutominoes, preprint, 2007.
%D A001700 M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed
animals, Discr. Math., 180 (1998), 73-106.
%D A001700 David Callan, A Combinatorial Interpretation for a Super-Catalan Recurrence,
Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.8.
%D A001700 H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed.,
1999, ch. V.3 (p. 122).
%D A001700 A. Frosini, R. Pinzani and S. Rinaldi, About half the middle binomial
coefficient, Pure Math. Appl., 11 (2000), 497-508.
%D A001700 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
%D A001700 M. D. McIlroy, personal communication.
%D A001700 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society
Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A001700 Phil J. Nahin, "An Imaginary Tale, The Story of [Sqrt(-1)]," Princeton
University Press, Princeton, NJ 1998, pg 62.
%D A001700 Problem 10753, Amer. Math. Monthly, 2000.
%H A001700 T. D. Noe, <a href="b001700.txt">Table of n, a(n) for n = 0..100</a>
%H A001700 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A001700 C. Borcea and I. Streinu, <a href="http://arXiv.org/abs/math.MG/0207126">
On the number of embeddings of minimally rigid graphs</a>.
%H A001700 M. Bousquet-M\'{e}lou, <a href="http://www.labri.fr/Perso/~bousquet/Articles/
Diriges/ani.ps.gz">New enumerative results on two-dimensional directed
animals</a>
%H A001700 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A001700 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</
a>
%H A001700 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=145">
Encyclopedia of Combinatorial Structures 145</a>
%H A001700 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
On generalizations of Stirling number triangles</a>, J. Integer Seqs.,
Vol. 3 (2000), #00.2.4.
%H A001700 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
The Hankel Transform and Some of its Properties</a>, J. Integer Sequences,
4 (2001), #01.1.5.
%H A001700 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Arithmetic and growth of periodic orbits</a>, J. Integer
Seqs., Vol. 4 (2001), #01.2.1.
%H A001700 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinomialCoefficient.html">Link to a section of The World of Mathematics.</
a>
%H A001700 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
OddGraph.html">Link to a section of The World of Mathematics.</a>
%H A001700 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/default.html">
Home Page</a>.
%H A001700 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder/Welcome.html">
(Old) Home Page</a>.
%H A001700 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001700 a(n-1) = binomial(2n, n)/2 = (2n)!/(2*n!*n!).
%F A001700 a(0) = 1, a(n) = 2(2n+1)a(n-1)/(n+1) for n > 0.
%F A001700 G.f.: (1/sqrt(1-4*x)-1)/(2*x).
%F A001700 Convolution of A000108 (Catalan) and A000984 (central binomial): Sum(C(k)*binomial(2*(n-k),
n-k), k=0..n), C(k) Catalan [ Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
]
%F A001700 a(n) = sum(k=0..n, C(n+k, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Aug 20 2002
%F A001700 a(n) = sum(k=0..n, C(n, k)*C(n+1, k+1)) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Oct 19 2002
%F A001700 a(n) = 4^n*binomial(n+1/2, n)/(n+1); - Paul Barry (pbarry(AT)wit.ie),
May 10 2005
%F A001700 E.g.f. Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)! = BesselI(1, 2x) . - Michael
Somos Jun 22 2005
%F A001700 E.g.f. in Maple notation: exp(2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)).
Integral representation as n-th moment of a positive function on
[0, 4]: a(n)=int(x^n*((x/(4-x))^(1/2)), x=0..4)/(2*Pi), n=0, 1...
This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr),
Oct 11 2001
%F A001700 Narayana transform of [1, 2, 3,...]. Let M = the Narayana triangle of
A001263 as an infinite lower triangular matrix and V = the Vector
[1, 2, 3,...]. Then A001700 = M * V. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Apr 25 2006
%F A001700 a(n) = C(2*n, n) + C(2*n, n-1); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jan 23 2007
%F A001700 a(n) = n*(n+1)*...*(2*n-1)/n! = product of n consecutive integers starting
with n, dividied by n!. - Jonathan Vos Post (jvospost3(AT)gmail.com),
Apr 09 2007
%F A001700 a(n):=(2n+1)*C(n); - Paul Barry (pbarry(AT)wit.ie), Aug 21 2007
%F A001700 Binomial transform of A005773 starting (1, 2, 5, 13, 35, 96,...) and
double binomial transform of A001405. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 01 2007
%F A001700 Row sums of triangle A132813. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 01 2007
%F A001700 Row sums of triangle A134285 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 19 2007
%F A001700 a(n) = (2n-1)!/(n!*(n-1)!) - William A. Tedeschi (fynmun(AT)hotmail.com),
Feb 27 2008
%F A001700 G.f.: F(1,3/2;2;4x). [From Paul Barry (pbarry(AT)wit.ie), Jan 23 2009]
%F A001700 G.f.: 1/(1-2x-x/(1-x/(1-x/(1-x/(1-... (continued fraction). [From Paul
Barry (pbarry(AT)wit.ie), May 06 2009]
%F A001700 G.f.: c(x)^2/(1-xc(x)^2), c(x) the g.f. of A000108. [From Paul Barry
(pbarry(AT)wit.ie), Sep 07 2009]
%p A001700 A001700 := n -> binomial(2*n+1,n+1);
%p A001700 seq((count(Composition(2*n),size=n+1)),n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 03 2007
%p A001700 with(combinat):seq(numbcomp(2*i,i), i=1..24) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 16 2007
%t A001700 Table[ Binomial[2n + 1, n + 1], {n, 0, 23} ]
%o A001700 (PARI) a(n)=binomial(2*n+1,n+1)
%Y A001700 Cf. A030662, A046097, A060897-A060900, A049027, A076025, A076026, A060150,
A028364, A050166, A039598, A001263, A005773, A001405, A132813, A134285.
%Y A001700 Equals A000984(n+1)/2. Cf. A030662, A046097. a(n)= (n+1)*Catalan(n) [A000108]
= A035324(n+1, 1) (first column of triangle).
%Y A001700 Row sums of triangles A028364, A050166, A039598.
%Y A001700 Bisections: a(2*k)= A002458(k), a(2*k+1)= A001448(k+1)/2, k>=0.
%Y A001700 Other versions of the same sequence: A088218, A110556, A138364.
%Y A001700 Diagonals 1 and 2 of triangle A100257.
%Y A001700 Second row of array A102539.
%Y A001700 Column of array A073165.
%Y A001700 Sequence in context: A122068 A099908 A167403 this_sequence A088218 A110556
A072266
%Y A001700 Adjacent sequences: A001697 A001698 A001699 this_sequence A001701 A001702
A001703
%K A001700 easy,nonn,nice,core
%O A001700 0,2
%A A001700 N. J. A. Sloane (njas(AT)research.att.com).
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