0,2

Number of arrangements of {1,2,...,n,n+1} containing the element 1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 11 2001

Comments from Thomas Wieder (wieder.thomas(AT)t-online.de), Oct 21 2004: "Also the number of hierarchies with unlabeled elements and labeled levels where the levels are permuted.

"Let l_x denote level x, e.g. l_2 is level 2. Let * denote an element. Then l_1*l_2***l_3** denotes a hierarchy of n=6 unlabeled elements with one element on level 1, three elements on level 2 and 2 elements on level 3.

"E.g. for n=3 one has a(3) = 11 possible hierarchies: l_1***, l_1**l_2*, l_1*l_2**, l_2**l_1*, l_2*l_1**, l_1*l_2*l_3*, l_3*l_1*l_2*, l_2*l_3*l_1*, l_1*l_3*l_2*, l_2*l_1*l_3*, l_3*l_2*l_1*. See A064618 for the number of hierarchies with labeled elements and labeled levels."

Polynomials in A010027 evaluated at 2.

Also the permanent of any n X n cofactor of an n+1 X n+1 version of J+I other than an n X n version of J+I (that is, a (1,2) matrix with n-1 2s, at most one per row and column). - D. G. Rogers, Aug 27 2006

a(n) = number of partitions of [n+1] into lists of sets that are both non-nesting and non-crossing. Non-nesting means that no set is contained in the span (interval from min to max) of another. For example, a(1) counts 12, 1-2, 2-1 and a(2) counts 123, 1-23, 23-1, 3-12, 12-3, 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, 3-2-1. - David Callan (callan(AT)stat.wisc.edu), Sep 20 2007

Row sums of triangle A137594. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 28 2008

Comments from Peter Bala (pbala(AT)toucansurf.com), Jul 10 2008 (Start): a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.

a(n) equals the sum of the lengths of the paths between a pair of distinct vertices of the complete graph K_(n+2) on n+2 vertices [Hassani]. For example, for the complete graph K_4 with vertex set {A,B,C,D} the 5 paths between A and B are AB of length 1, ACB and ADB, both of length 2 and ACDB and ADCB, both of length 3. The sum of the lengths is 1+2+2+3+3 = 11 = a(2).

The number of paths between 2 distinct vertices of K_n is equal to A000522(n-2); the number of simple cycles through a vertex of K_n equals A038154(n-1).

Recurrence relation: a(0) = 1, a(1) = 3, a(n) = (n+2)*a(n-1) - (n-1)*a(n-2) for n >=2. The sequence b(n) := n*n! = A001563(n) satisfies the same recurrence with the initial conditions b(0) = 0, b(1) = 1. This leads to the finite continued fraction expansion a(n)/b(n) = 3-1/(4-2/(5-3/(6-...-(n-1)/(n+2)))), n >= 1.

Lim n -> infinity a(n)/b(n) = e = 3-1/(4-2/(5-3/(6-...-n/((n+3)-...)))).

For n >= 1, a(n) = b(n)*(3 - sum {k = 2..n} 1/(k!*(k-1)*k) (see the remark by Deutsch above) since the rhs satisfies the above recurrence with the same initial conditions. Hence e = 3 - sum {k = 2..inf} 1/(k!*(k-1)*k).

For sequences satisfying the more general recurrence a(n) = (n+1+r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n;-1), refer to A000522 (r=0), A082030 (r=2), A095000 (r=3) and A095177 (r=4). (End)

Binomial transform of n! Offset 1. a(3)=11. [From Al Hakanson (hawkuu(AT)gmail.com), May 18 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas. Freeman, San Francisco, 1974.

Biondi, E.; Divieti, L.; Guardabassi, G.; Counting paths, circuits, chains and cycles in graphs: A unified approach. Canad. J. Math. 22 1970 22-35.

M. J. Knight and W. O. Egerland, Solution to Problem 5911, Amer. Math. Monthly 81 (1974) 675-676.

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 56, ex. 232.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 400

F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras

M. Hassani, Counting and computing by e

Weisstein, Eric W., Poisson-Charlier polynomial

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

E.g.f.: exp(x)*(1-x)^(-2). a(n) = round(evalf(exp(1)*(n-1)*(n-1)!)) (n>1).

a(n) = floor(n*n!*e)+1 - Melvin J. Knight (knightmj(AT)juno.com), May 30 2001

The n-th row of array A089900 is the n-th binomial transform of this sequence. The (n+1)-th term of the n-th binomial transform is (n+1)^(n+1), for n>=0. E.g. the 5-th term of the 4-th binomial transform is 5^5: [1, 7, 51, 389, 3125, ..]. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2003

G.f.: Sum_{k>=0} k!*(x/(1-x))^k. - Michael Somos Mar 04 2004

a(n) = Sum_{k = 0..n} A046716(n, k)*2^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 12 2004

(n-1)*a(n) = n^2*a(n-1)-1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 04 2004

a(n) = Sum[P(n, k)*(k+1), {k, 0, n}]. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 28 2005

a(n) = n!*n*[3 - Sum(1/(j(j-1)j!,j=2..n)] for n>=1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 12 2008

a(2) = 11: {1,12,21,13,31,123,132,213,231,312,321}

a:=proc(n) options operator, arrow: factorial(n)*n*(3-(sum(1/(j*(j-1)*factorial(j)), j=2..n))) end proc: 1, seq(a(n), n=1..20); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 12 2008

(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(x+x*O(x^n))/(1-x)^2, n))

Cf. A001563, A089900, A064618, A137594.

a(n)=A000522(n+1)-A000522(n).

First differences of A000522, A007526, A026243, A073591. Equals (1/2) A006183(n-2).

Equals A036918(n+1) + 1.

Sequence in context: A025539 A074528 A004211 this_sequence A012316 A058733 A024333

Adjacent sequences: A001336 A001337 A001338 this_sequence A001340 A001341 A001342

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Typo in description in 1995 Encyc. Int. Seqs. corrected Mar 15, 1997.