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A112029 a(n) = Sum_{k=0..n} binomial(n+k, k)^2. 9
1, 5, 46, 517, 6376, 82994, 1119210, 15475205, 217994860, 3115374880, 45035696036, 657153097330, 9663914317396, 143050882063262, 2129448324373546, 31853280798384645, 478503774600509620, 7215090439396842572, 109154411037070011504, 1656268648035559711392 (list; graph; refs; listen; history; text; internal format)
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.

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

Pedro J. Miana, Hideyuki Ohtsuka, 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

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 *)

CROSSREFS

Cf. A001700, A112028, A219562, A219563, A219564

Sequence in context: A198256 A232972 A127304 * A295544 A058478 A159608

Adjacent sequences:  A112026 A112027 A112028 * A112030 A112031 A112032

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 28 2005

STATUS

approved

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Last modified October 31 05:06 EDT 2020. Contains 338098 sequences. (Running on oeis4.)