This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014178 #27 Jul 08 2025 02:54:27
%S A014178 1,3,31,399,5871,93753,1577479,27556623,495001327,9085988613,
%T A014178 169675769781,3213444254133,61573700137431,1191526503165729,
%U A014178 23252920338835911,457112339182896399,9043566887755775727,179928068420530483389,3597714616543167088921
%N A014178 a(n) = Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k).
%H A014178 Vincenzo Librandi, <a href="/A014178/b014178.txt">Table of n, a(n) for n = 0..200</a>
%H A014178 C. Elsner, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulas for sums involving binomial coefficients</a>, Fib. Q., 43,1 (2005), 31-45.
%H A014178 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012
%F A014178 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
%F A014178 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
%F A014178 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
%p A014178 f := n->sum( 'binomial(n,k)^3*binomial(n+k,k)^1','k'=0..n);
%t A014178 Table[Sum[Binomial[n,k]^3*Binomial[n+k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Nov 04 2012 *)
%o A014178 (PARI) a(n)=sum(k=0,n, binomial(n,k)^3*binomial(n+k,k) ); \\ _Joerg Arndt_, May 04 2013
%Y A014178 Cf. A218693, A112019, A111968, A014180, A218689, A218692, A005258, A005259.
%K A014178 nonn,easy
%O A014178 0,2
%A A014178 _N. J. A. Sloane_