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

%I A112028 #46 Jan 05 2025 19:51:38
%S A112028 1,9,244,9065,389376,18188478,897376152,46011772521,2427553965160,
%T A112028 130930630643384,7186614533569296,400132290102421214,
%U A112028 22543708920891189136,1282873288801683197250,73628947696550668509744,4257138240245923453355625,247733479854085081062353400
%N A112028 a(n) = Sum_{k=0..n} binomial(n+k,k)^3.
%D A112028 M.J. Coster: Supercongruences, [Thesis] Univ. of Leiden, the Netherlands, 1988.
%H A112028 Vincenzo Librandi, <a href="/A112028/b112028.txt">Table of n, a(n) for n = 0..200</a>
%H A112028 F. Baldassarri, S. Bosch, B. Dwork, (eds), <a href="https://doi.org/10.1007/BFb0091139">p-adic Analysis. Lecture Notes in Mathematics</a>, vol. 1454, pp. 194 - 204,  Springer, Berlin, Heidelberg.
%H A112028 Matthijs Coster, <a href="https://core.ac.uk/download/pdf/301642554.pdf">Supercongruences</a>.
%H A112028 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 A112028 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
%F A112028 a(n) ~ 2^(6*n+3)/(7*(Pi*n)^(3/2)). - _Vaclav Kotesovec_, Nov 23 2012
%F A112028 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
%F A112028 O.g.f. can be expressed in terms of hypergeometric functions (see Maple program). - _Mark van Hoeij_, Apr 01 2013
%F A112028 From _Peter Bala_, Mar 29 2023: (Start)
%F A112028 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.
%F A112028 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)
%p A112028 y2 := hypergeom([2/3, 2/3],[4/3],-x^2/27)*x^(1/3);
%p A112028 h := hypergeom([1/4, 1/4],[1],64*x)^2;
%p A112028 H := (243+247*x)*x*diff(h,x,x) + (463*x+234)*diff(h,x) + (48-24/x)*h;
%p A112028 ogf := y2*Int(Int(y2*H,x)/(8*x*(x^2+27)*y2^2),x);
%p A112028 # Check ogf by computing a series expansion:
%p A112028 SER := proc(a, x) series(a, x, 20) end:
%p A112028 INT := proc(a, x) int(SER(a, x), x) end:
%p A112028 SER(eval(ogf, Int = INT), x); # _Mark van Hoeij_, Apr 04 2013
%t A112028 Table[Sum[Binomial[n+k,k]^3,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Sep 24 2012 *)
%o A112028 (PARI) a(n) = sum(k=0, n, binomial(n+k, k)^3); \\ _Michel Marcus_, Mar 09 2016
%Y A112028 Cf. A001700, A112029, A219562, A219563, A219564.
%Y A112028 A375178 is an essentially identical sequence.
%K A112028 nonn,easy
%O A112028 0,2
%A A112028 _N. J. A. Sloane_, Nov 28 2005