A089662 a(n) = S1(n,5), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

0, 2, 131, 2172, 20386, 138580, 763824, 3631712, 15470144, 60527232, 221297920, 765580288, 2529498624, 8039103488, 24713744384, 73818562560, 215011065856, 612515381248, 1710842904576, 4695105732608, 12682107944960, 33768108982272, 88748191645696

Offset

0,2

Links

Vincenzo Librandi, 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.

Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Formula

There is an explicit formula for the sum - see Wang and Zhang, p. 334.

From Chai Wah Wu, Jun 21 2016: (Start)

a(n) = 14*a(n-1) - 84*a(n-2) + 280*a(n-3) - 560*a(n-4) + 672*a(n-5) - 448*a(n-6) + 128*a(n-7) for n > 6.

G.f.: x*(-16*x^5 + 64*x^4 + 422*x^3 + 506*x^2 + 103*x + 2)/(1 - 2*x)^7. (End)

a(n) = 2^(n-7)*n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4). - Ilya Gutkovskiy, Jun 21 2016

E.g.f.: (1/4)*x*(8 + 246*x + 940*x^2 + 1015*x^3 + 376*x^4 + 42*x^5)*exp(2*x). - G. C. Greubel, May 24 2022

Mathematica

LinearRecurrence[{14, -84, 280, -560, 672, -448, 128}, {0, 2, 131, 2172, 20386, 138580, 763824}, 30] (* Vincenzo Librandi, Jun 22 2016 *)

Prog

(Magma) [2^(n-7)*n*(21*n^5+61*n^4+55*n^3+15*n^2-28*n+4): n in [0..30]]; // Vincenzo Librandi, Jun 22 2016
(SageMath) [2^(n-7)*n*(21*n^5 +61*n^4 +55*n^3 +15*n^2 -28*n +4) for n in (0..40)] # G. C. Greubel, May 24 2022

Crossrefs

Sequences of S1(n,t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), this sequence (t=5), A089663 (t=6).

Sequence in context: A084549 A142251 A125633 * A119778 A071606 A080282

Adjacent sequences: A089659 A089660 A089661 * A089663 A089664 A089665

Keyword

nonn

Author

N. J. A. Sloane, Jan 04 2004

Status

approved