OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
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
Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.
FORMULA
a(n) ~ 2^(4*n+2)/(3*Pi*n). - Vaclav Kotesovec, Nov 23 2012
Recurrence: 2*(2*n+1)*(21*n-13)*n^2*a(n) = (1365*n^4 - 1517*n^3 + 240*n^2 + 216*n - 64)*a(n-1) - 4*(n-1)*(2*n-1)^2*(21*n+8)*a(n-2). - Vaclav Kotesovec, Nov 23 2012
G.f.: see Maple code. - Mark van Hoeij, Mar 27 2013
a(p-1) == 1 (mod p^3) for all primes p >= 5. See the comments in A173774. - Peter Bala, Jul 12 2024
a(n-1) = 1/(4*n) * binomial(2*n, n)^2 * ( 1 + 3*((n - 1)/(n + 1))^3 + 5*((n - 1)*(n - 2)/((n + 1)*(n + 2)))^3 + 7*((n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)))^3 + ... ) for n >= 1. - Peter Bala, Jul 22 2024
MAPLE
f := 64*x^2/(16*x-1); S := sqrt(x)*sqrt(4-x);
H := ((10*x-5/8)*hypergeom([1/4, 1/4], [1], f)-(21*x-21/8)*hypergeom([1/4, 5/4], [1], f))/(S*(1-16*x)^(5/4));
ord := 30;
ogf := series(int(series(H, x=0, ord), x)/S, x=0, ord);
# Mark van Hoeij, Mar 27 2013
MATHEMATICA
Table[Sum[Binomial[n+k, k]^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 23 2012 *)
PROG
(Magma) [(&+[Binomial(n+j, j)^2: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 06 2021
(Sage) [sum(binomial(n+j, j)^2 for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 06 2021
(PARI) a(n) = sum(k=0, n, binomial(n+k, k)^2); \\ Michel Marcus, Jul 07 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2005
STATUS
approved