A112029 a(n) = Sum_{k=0..n} binomial(n+k, k)^2.

1, 5, 46, 517, 6376, 82994, 1119210, 15475205, 217994860, 3115374880, 45035696036, 657153097330, 9663914317396, 143050882063262, 2129448324373546, 31853280798384645, 478503774600509620, 7215090439396842572, 109154411037070011504, 1656268648035559711392

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

Cf. A001700, A112028, A173774, A219562, A219563, A219564, A333592.

Sequence in context: A198256 A232972 A127304 * A371380 A295544 A058478

Adjacent sequences: A112026 A112027 A112028 * A112030 A112031 A112032

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 28 2005

Status

approved