A089665 a(n) = S2(n,2), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

0, 4, 73, 788, 6630, 48120, 316526, 1940568, 11284380, 62968560, 339954670, 1786320184, 9176663028, 46248446608, 229285525420, 1120646918000, 5409322603896, 25824570392544, 122086747617198, 572130452101240, 2660063893120900, 12279619924999504, 56318986959592676

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Formula

a(n) = (1/24)*n*( (n+1)*(7*n+5)*4^n - 2*(n-1)*(3*n^2 - 2*n + 1)*binomial(2*n, n)/(2*n-1) ). (See Wang and Zhang, p. 338.)

From G. C. Greubel, May 25 2022: (Start)

a(n) = (n/6)*( (n+1)*(7*n+5)*4^(n-1) - (n-1)*(3*n^2 - 2*n + 1)*Catalan(n-1) ).

G.f.: x*(4*(1+3*x) - x*(3 + 2*x + 4*x^2)*sqrt(1-4*x))/(1-4*x)^4.

E.g.f.: x*(4 + 22*x + 56*x^2/3)*exp(4*x) + (x^2/6)*exp(2*x)*( -(9 + 62*x + 145*x^2 + 84*x^3)*f(x, 0) + (36 + 99*x - 32*x^2 - 84 x^3)*f(x, 1) + (45 + 270*x + 284*x^2 + 48*x^3)*f(x, 2) + x*(109 + 224*x + 78*x^2)*f(x, 3) + x^2*(53 + 36*x)*f(x, 4) + 6*x^3*f(x, 5) ), where f(x, n) = BesselI(n, 2*x). (End)

Maple

S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);
seq(S2(n, 2), n = 0..40);

Mathematica

Table[(1/24)*(n*(n+1)*(7*n+5)*4^n -4*(n-1)*(3*n^2-2*n+1)*Binomial[2*n-2, n-1]), {n, 0, 40}] (* G. C. Greubel, May 25 2022 *)

Prog

(SageMath) [(n/6)*((n+1)*(7*n+5)*4^(n-1) -(n-1)*(3*n^2-2*n+1)*catalan_number(n-1)) for n in (0..40)] # G. C. Greubel, May 25 2022

Crossrefs

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), this sequence (t=2), A089666 (t=3), A089667 (t=4), A089668 (t=5).

Cf. A000108, A089658, A089669.

Sequence in context: A284673 A055556 A168299 * A092871 A222767 A353740

Adjacent sequences: A089662 A089663 A089664 * A089666 A089667 A089668

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 04 2004

Status

approved