A092814 - OEIS

%I #26 Oct 05 2022 13:08:19

%S 1,17,2593,990737,473940001,261852948017,166225611652129,

%T 115586046457265681,85467827222155042849,66421846251482628852017,

%U 53755021948680412765238593,44947131400352317819689905201,38613445585740736549461528111649,33942058336306457714420306982430001

%N Schmidt's problem sum for r = 4.

%H Vincenzo Librandi, <a href="/A092814/b092814.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>

%F a(n) = sum(k=0..n, binomial(n,k)^4 * binomial(n+k,k)^4 ).

%F a(n) ~ (1+sqrt(2))^(4*(2n+1))/(2^(15/4)*(Pi*n)^(7/2)). - _Vaclav Kotesovec_, Nov 04 2012

%t Table[Sum[Binomial[n, k]^4 Binomial[n+k, k]^4, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 04 2012 *)

%t a[k_]:= HypergeometricPFQ[{-k,-k,-k,-k,1+k,1+k,1+k,1+k}, {1,1,1,1,1,1,1}, 1]

%t Table[ a[k], {k, 0, 20}] (* _Gerry Martens_, Sep 26 2022 *)

%o (PARI) a(n)=sum(k=0,n,binomial(n,k)^4*binomial(n+k,k)^4); \\ _Joerg Arndt_, May 11 2013

%Y Cf. A001850, A005259, A092813, A092815, A218689.

%K nonn

%O 0,2

%A _Eric W. Weisstein_, Mar 06 2004

%E Prepended missing a(0)=1, _Joerg Arndt_, May 11 2013