Search: id:A003506
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%I A003506
%S A003506 1,2,2,3,6,3,4,12,12,4,5,20,30,20,5,6,30,60,60,30,6,7,42,105,140,105,42,
%T A003506 7,8,56,168,280,280,168,56,8,9,72,252,504,630,504,252,72,9,10,90,360,840,
%U A003506 1260,1260,840,360,90,10,11,110,495,1320,2310,2772,2310,1320,495,110,11
%N A003506 Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 
               1, 1<=k<=n.
%C A003506 Array 1/Beta(n,m) read by antidiagonals. - Michael Somos Feb 05 2004
%C A003506 a(n,3) = A027480(n-2); a(n,4) = A033488(n-3). - Ross La Haye (rlahaye(AT)new.rr.com), 
               Feb 13 2004
%C A003506 a(n,k) = total size of all of the elements of the family of k-size subsets 
               of an n-element set. For example, a 2-element set, say, {1,2}, has 
               3 families of k-size subsets: one with 1 0-size element, one with 
               2 1-size elements and one with 1 2-size element; respectively, {{}}, 
               {{1},{2}}, {{1,2}}. - Ross La Haye (rlahaye(AT)new.rr.com), Dec 31 
               2006
%C A003506 Second slice along the 1-2-plane in the cube a(m,n,o) = a(m-1,n,o)+a(m,
               n-1,o)+a(m,n,o-1) with a(1,0,0)=1 and a(m<>1=0,n>=0,0>=o)=0, for 
               which the first slice is Pascal's triangle (slice read by anti-diagonals). 
               - Thomas Wieder (thomas.wieder(AT)t-online.de), Aug 06 2006
%C A003506 Triangle, read by rows, given by [2,-1/2,1/2,0,0,0,0,0,0,...] DELTA [2,
               -1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in 
               A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 07 2007
%C A003506 This sequence * [1/1, 1/2, 1/3,...] = (1, 3, 7, 15, 31,...) - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Nov 14 2007
%C A003506 n-th row = coefficients of first derivative of corresponding Pascal's 
               triangle row. Example: x^4 + 4x^3 + 6x^2 + 4x + 1 becomes (4, 12, 
               12, 4). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2007
%D A003506 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, see 130.
%D A003506 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), 
               FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published 
               by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; 
               see p. 38.
%D A003506 G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, 
               p. 26.
%D A003506 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
%D A003506 M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial 
               equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, 
               eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. 
               See p. 152.
%D A003506 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. 
               Penguin Books, NY, 1986, 35.
%H A003506 D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">
               Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%H A003506 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/">
               Home Page</a>.
%H A003506 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder">(Old) 
               Home Page</a>.
%H A003506 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LeibnizHarmonicTriangle.html">Link to a section of The World of Mathematics.</
               a>
%F A003506 a(n, 1)=1/n; a(n, k)=a(n-1, k-1)-a(n, k-1) for k>1.
%F A003506 Considering the integer values (rather than unit fractions): a(n, k) 
               =k*C(n, k) =n*C(n-1, k-1) =a(n, k-1)*a(n-1, k-1)/(a(n, k-1)-a(n-1, 
               k-1)) =a(n-1, k)+a(n-1, k-1)*k/(k-1) =(a(n-1, k)+a(n-1, k-1))*n/(n-1) 
               =k*A007318(n, k) =n*A007318(n-1, k-1). Row sums of integers are n*2^(n-1)=A001787(n); 
               row sums of the unit fractions are A003149(n-1)/A000142(n). - Henry 
               Bottomley (se16(AT)btinternet.com), Jul 22 2002
%F A003506 G.f.: x*y/(1-x-y*x)^2. E.g.f: x*y*exp(x+x*y). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Nov 01 2003
%F A003506 T(n,k) = n*binomial(n-1,k-1)= n*A007318(n-1,k-1) . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Aug 04 2006
%F A003506 Binomial transform of A128064(unsigned). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 29 2007
%F A003506 t(n,m)=Gamma[n]/(Gamma[n - m]*Gamma[m]. - Roger L. Bagula and Gary W. 
               Adamson (rlbagulatftn(AT)yahoo.com), Sep 14 2008
%F A003506 f[s,n]=Integrate[Exp[ -s*x]*x^n,{x,0,Infinity}]=Gamma[n]/s^n; t(n,m)=f[s,
               n]/(f[s,n-m]*f[s,m])=Gamma[n]/(Gamma[n - m]*Gamma[m]; the powers 
               of s cancel out. - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Sep 14 2008
%e A003506 1/1; 1/2, 1/2; 1/3, 1/6, 1/3; 1/4, 1/12, 1/12, 1/4; 1/5, 1/20, 1/30, 
               1/20, 1/5; ...
%e A003506 Triangle begins:
%e A003506 {1},
%e A003506 {2, 2},
%e A003506 {3, 6, 3},
%e A003506 {4, 12, 12, 4},
%e A003506 {5, 20, 30, 20, 5},
%e A003506 {6, 30, 60, 60, 30, 6},
%e A003506 {7, 42, 105, 140, 105, 42, 7},
%e A003506 {8, 56, 168, 280, 280, 168, 56, 8},
%e A003506 {9, 72, 252, 504, 630, 504, 252, 72, 9},
%e A003506 {10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10},
%e A003506 {11, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 11}
%p A003506 with(combstruct):for n from 0 to 11 do seq(m*count(Combination(n), size=m), 
               m = 1 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 09 2008
%t A003506 L[n_, 1] := 1/n; L[n_, m_] := L[n, m] = L[n - 1, m - 1] - L[n, m - 1]; 
               Take[ Flatten[ Table[ 1 / L[n, m], {n, 1, 12}, {m, 1, n}]], 66]
%t A003506 t[n_, m_] = Gamma[n]/(Gamma[n - m]*Gamma[m]); Table[Table[t[n, m], {m, 
               1, n - 1}], {n, 2, 12}]; Flatten[%] - Roger L. Bagula and Gary W. 
               Adamson (rlbagulatftn(AT)yahoo.com), Sep 14 2008
%o A003506 (PARI) A(i,j)=if(i<1|j<1,0,1/subst(intformal(x^(i-1)*(1-x)^(j-1)),x,1))
%o A003506 (PARI) A(i,j)=if(i<1|j<1,0,1/sum(k=0,i-1,(-1)^k*binomial(i-1,k)/(j+k)))
%Y A003506 Cf. A007622, A128064.
%Y A003506 Cf. A094305, A121547, A121306, A119800, A002378, A007318.
%Y A003506 Row sums are in A001787. Central column is A002457. Half-diagonal is 
               in A090816.
%Y A003506 Sequence in context: A051173 A128228 A125102 this_sequence A047662 A075196 
               A015050
%Y A003506 Adjacent sequences: A003503 A003504 A003505 this_sequence A003507 A003508 
               A003509
%K A003506 tabl,nonn,nice,easy
%O A003506 1,2
%A A003506 N. J. A. Sloane (njas(AT)research.att.com).
%E A003506 Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 07 2007

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