A014178 a(n) = Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k).
1, 3, 31, 399, 5871, 93753, 1577479, 27556623, 495001327, 9085988613, 169675769781, 3213444254133, 61573700137431, 1191526503165729, 23252920338835911, 457112339182896399, 9043566887755775727, 179928068420530483389, 3597714616543167088921
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
- V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012
Programs
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Maple
f := n->sum( 'binomial(n,k)^3*binomial(n+k,k)^1','k'=0..n);
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Mathematica
Table[Sum[Binomial[n,k]^3*Binomial[n+k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 04 2012 *) -
PARI
a(n)=sum(k=0,n, binomial(n,k)^3*binomial(n+k,k) ); \\ Joerg Arndt, May 04 2013
Formula
a(n) ~ (1+r)^(2*n+3/2)/r^(4*n+7/2)/(4*Pi^(3/2)*n^(3/2))*sqrt((1-r)/(2+r)), where r is positive real root of the equation (1-r)^3*(1+r)=r^4, r = 0.58252220781047... - Vaclav Kotesovec, Nov 04 2012
Recurrence: (n-1)*n^3*(29412*n^4 - 246240*n^3 + 764259*n^2 - 1042332*n + 527381)*a(n) = 2*(n-1)*(235296*n^7 - 2322864*n^6 + 9245766*n^5 - 19022421*n^4 + 21621181*n^3 - 13561627*n^2 + 4459053*n - 605664)*a(n-1) + 2*(1647072*n^8 - 20377728*n^7 + 107506956*n^6 - 315721020*n^5 + 564159163*n^4 - 627527310*n^3 + 423779896*n^2 - 158592459*n + 25128864)*a(n-2) + 2*(n-2)*(4176504*n^7 - 49583844*n^6 + 243933522*n^5 - 641841009*n^4 + 971188553*n^3 - 841622632*n^2 + 385567572*n - 72023040)*a(n-3) - 4*(n-2)*(29412*n^4 - 128592*n^3 + 202011*n^2 - 134886*n + 32480)*(n-3)^3*a(n-4). - Vaclav Kotesovec, Nov 04 2012
The expansions exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 20*x^2 + 184*x^3 + 2060*x^4 + 26246*x^5 + ... and exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + x + 2*x^2 + 12*x^3 + 112*x^4 + 1296*x^5 + ... appear to have integer coefficients. Cf. A005258 and A005259. - Peter Bala, Jan 14 2016