A092814 Schmidt's problem sum for r = 4.

1, 17, 2593, 990737, 473940001, 261852948017, 166225611652129, 115586046457265681, 85467827222155042849, 66421846251482628852017, 53755021948680412765238593, 44947131400352317819689905201, 38613445585740736549461528111649, 33942058336306457714420306982430001

Offset: 0

Author

Eric W. Weisstein, Mar 06 2004

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012
• Eric Weisstein's World of Mathematics, Schmidt's Problem

Crossrefs

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

Programs

Mathematica

Table[Sum[Binomial[n, k]^4 Binomial[n+k, k]^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 04 2012 *)
a[k_]:= HypergeometricPFQ[{-k,-k,-k,-k,1+k,1+k,1+k,1+k}, {1,1,1,1,1,1,1}, 1]
Table[ a[k], {k, 0, 20}] (* Gerry Martens, Sep 26 2022 *)

PARI

a(n)=sum(k=0,n,binomial(n,k)^4*binomial(n+k,k)^4); \\ Joerg Arndt, May 11 2013

Formula

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

a(n) ~ (1+sqrt(2))^(4*(2n+1))/(2^(15/4)*(Pi*n)^(7/2)). - Vaclav Kotesovec, Nov 04 2012

Extensions

Prepended missing a(0)=1, Joerg Arndt, May 11 2013