A089663 a(n) = S1(n, 6), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

0, 2, 259, 6284, 77180, 646960, 4235864, 23313408, 112793088, 493969920, 1998346240, 7577934848, 27232132096, 93517705216, 308908943360, 986642513920, 3059995508736, 9247515082752, 27310549696512, 79012328898560, 224396746424320, 626707269681152

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.

Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Formula

a(n) = (1/21)*n*(n+1)*(381*n^5 + 921*n^4 + 381*n^3 + 39*n^2 - 746*n + 368) * 2^(n-7). (See Wang and Zhang, p. 334.)

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

a(n) = 16*a(n-1) - 112*a(n-2) + 448*a(n-3) - 1120*a(n-4) + 1792*a(n-5) - 1792*a(n-6) + 1024*a(n-7) - 256*a(n-8) for n > 7.

G.f.: x*(2 + 227*x + 2364*x^2 + 4748*x^3 + 2096*x^4 - 72*x^5)/(1 - 2*x)^8. (End)

a(n) = 2^(n-7)*n*(381*n^6 + 1302*n^5 + 1302*n^4 + 420*n^3 - 707*n^2 - 378*n + 368)/21. - Ilya Gutkovskiy, Jun 21 2016

E.g.f.: (1/42)*x*(84 + 5271*x + 33278*x^2 + 57855*x^3 + 37086*x^4 + 9303*x^5 +

762*x^6)*exp(2*x). - G. C. Greubel, May 24 2022

Mathematica

LinearRecurrence[{16, -112, 448, -1120, 1792, -1792, 1024, -256}, {0, 2, 259, 6284, 77180, 646960, 4235864, 23313408}, 40] (* G. C. Greubel, Jun 22 2016 *)

Prog

(Magma) [2^(n-7)*n*(381*n^6+1302*n^5+1302*n^4+420*n^3-707*n^2-378*n+368)/21: n in [0..40]]; // G. C. Greubel, May 24 2022
(SageMath) [n*(n+1)*(381*n^5 +921*n^4 +381*n^3 +39*n^2 -746*n +368)*2^(n-7)/21 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), A089662 (t=5), this sequence (t=6).

Cf. A089664, A089669.

Sequence in context: A128697 A182422 A218435 * A252708 A103029 A122862

Adjacent sequences: A089660 A089661 A089662 * A089664 A089665 A089666

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 04 2004

Status

approved