OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
F. Baldassarri, S. Bosch, B. Dwork, (eds), p-adic Analysis. Lecture Notes in Mathematics, vol. 1454, pp. 194 - 204, Springer, Berlin, Heidelberg.
Matthijs Coster, Supercongruences.
Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012
FORMULA
a(n) ~ 2^(8*n+4)/(15*Pi^2*n^2).
Recurrence: 4*(n-1)*(4*n-1)*(4*n+1)*(279825*n^6 - 2240985*n^5 + 7416081*n^4 - 12962383*n^3 + 12597634*n^2 - 6438500*n + 1347304)*n^4*a(n) = 2*(n-1)*(2290647450*n^12 - 22926837585*n^11 + 100717526436*n^10 - 254986993727*n^9 + 410380920831*n^8 - 435959897978*n^7 + 305660392723*n^6 - 134977315842*n^5 + 31413259700*n^4 + 2833672*n^3 - 2076143616*n^2 + 500898816*n - 39813120)*a(n-1) + (859902225*n^13 - 10755967005*n^12 + 60090860763*n^11 - 197381561581*n^10 + 422055067481*n^9 - 613861172995*n^8 + 615013106513*n^7 - 418396400175*n^6 + 182810864162*n^5 - 42759392772*n^4 + 146171272*n^3 + 2813432832*n^2 - 691172352*n + 55738368)*a(n-2) - 16*(n-2)^3*(2*n-3)^4*(279825*n^6 - 562035*n^5 + 408531*n^4 - 111409*n^3 - 5504*n^2 + 7968*n - 1024)*a(n-3).
G.f. as an expression in terms of 2F1 and 4F3 functions is given in the Maple program below. - Mark van Hoeij, Apr 02 2013
From Peter Bala, Nov 29 2024: (Start)
Conjecture: a(p-1) == 1 (mod p^5) for prime p >= 7 (checked up to p = 499). Coster, Theorem 4, proves that a(p-1) == 1 (mod p^3) for primes p >= 5.
Conjecture: 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 >= 5. Coster, Theorem 4, proves that a(p^r -1) == a(p^(r-1) - 1) (mod p^(3*r)) for r >= 2 and all primes p >= 5. (End)
MAPLE
q := x-4+I*((x+4)*(16-x))^(1/2);
f := x*(q/8)^4;
s := ((q-2)/(8*I-6))^(1/4);
y1 := hypergeom([1/8, 1/8], [3/4], f) * s / x^(1/8);
r := 2/(x*((x+4)*(16-x))^(1/2)*y1^2);
h := hypergeom([1/2, 1/2, 1/2, 1/2], [1, 1, 1], 256*x);
u := (15*(223*x+72)*x^2*diff(h, x, x, x)+(14579*x+3226)*x*diff(h, x, x)
+(9969*x+1002)*diff(h, x)+320*h)/(16*(16-x)*(x+4)*x^2);
ogf := y1^2*Int(r*(1+Int(r*Int(u/(r*y1)^2, x), x)), x) ;
# Check o.g.f. 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 02 2013
MATHEMATICA
Table[Sum[Binomial[n+k, k]^4, {k, 0, n}], {n, 0, 20}]
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+k, k)^4); \\ Michel Marcus, Jul 15 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Nov 23 2012
STATUS
approved