A112028 a(n) = Sum_{k=0..n} binomial(n+k,k)^3.

1, 9, 244, 9065, 389376, 18188478, 897376152, 46011772521, 2427553965160, 130930630643384, 7186614533569296, 400132290102421214, 22543708920891189136, 1282873288801683197250, 73628947696550668509744, 4257138240245923453355625, 247733479854085081062353400

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Matthijs Coster, Supercongruences, Ph.D. thesis, Universiteit Leiden, Jun 08, 1988.

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Formula

a(n) ~ 2^(6*n+3)/(7*(Pi*n)^(3/2)). - Vaclav Kotesovec, Nov 23 2012

Recurrence: 3*(3*n-1)*(3*n+1)*(15799*n^5 - 103177*n^4 + 265789*n^3 - 336367*n^2 + 208000*n - 49852)*n^3*a(n) = 24*(2*n-3)^3*(3*n-4)*(3*n-2)*(15799*n^5 - 24182*n^4 + 11071*n^3 - 72*n^2 - 1080*n + 192)*a(n-1) - (n-1)^2*(15799*n^5 - 103177*n^4 + 265789*n^3 - 336367*n^2 + 208000*n - 49852)*n^3*a(n-2) + 8*(n-2)^2*(2*n-3)^3*(15799*n^5 - 24182*n^4 + 11071*n^3 - 72*n^2 - 1080*n + 192)*a(n-3). - Vaclav Kotesovec, Nov 23 2012

O.g.f. can be expressed in terms of hypergeometric functions (see Maple program). - Mark van Hoeij, Apr 01 2013

From Peter Bala, Mar 29 2023: (Start)

The supercongruence a(p-1) == 1 (mod p^5) appears to hold for all primes p >= 7 (checked up to p = 199). Coster, Theorem 4, proves that a(p-1) == 1 (mod p^3) for primes p >= 5.

For r >= 2, the supercongruence a(p^r - 1) == a(p^(r-1) - 1) (mod p^(3*r+3)) may hold for all primes p >= 7. (End)

Maple

y2 := hypergeom([2/3, 2/3], [4/3], -x^2/27)*x^(1/3);
h := hypergeom([1/4, 1/4], [1], 64*x)^2;
H := (243+247*x)*x*diff(h, x, x) + (463*x+234)*diff(h, x) + (48-24/x)*h;
ogf := y2*Int(Int(y2*H, x)/(8*x*(x^2+27)*y2^2), x);
# Check ogf by computing a series expansion:
SER := proc(a, x) series(a, x, 20) end:
INT := proc(a, x) int(SER(a, x), x) end:
SER(eval(ogf, Int = INT), x); # Mark van Hoeij, Apr 04 2013

Mathematica

Table[Sum[Binomial[n+k, k]^3, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Sep 24 2012 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+k, k)^3); \\ Michel Marcus, Mar 09 2016

Crossrefs

Cf. A001700, A112029, A219562, A219563, A219564.

Sequence in context: A295228 A217628 A167005 * A368769 A329305 A183235

Adjacent sequences: A112025 A112026 A112027 * A112029 A112030 A112031

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 28 2005

Status

approved