A089660 a(n) = S1(n,3), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

0, 2, 35, 276, 1522, 6820, 26664, 94640, 312512, 975744, 2913280, 8386048, 23416320, 63724544, 169637888, 443043840, 1137934336, 2879979520, 7194083328, 17761304576, 43390730240, 104997322752, 251881062400, 599482433536, 1416470986752, 3324615065600

Offset: 0

Author

N. J. A. Sloane, Jan 04 2004

Keywords

• nonn
• easy

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 (10,-40,80,-80,32).

Crossrefs

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), A089659 (t=2), this sequence (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

Programs

Magma

[2^(n-6)*n*(3*n*(7+10*n+5*n^2) -2): n in [0..40]]; // G. C. Greubel, May 24 2022

Mathematica

LinearRecurrence[{10,-40,80,-80,32}, {0,2,35,276,1522}, 40] (* Vincenzo Librandi, Jun 22 2016 *)

SageMath

[n*(15*n^3+30*n^2+21*n-2)*2^(n-6) for n in (0..40)] # G. C. Greubel, May 24 2022

Formula

a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*2^(n-6). - R. J. Mathar, Sep 16 2009

G.f.: x*(2 + 15*x + 6*x^2 + 2*x^3)/(1-2*x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009

a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5) for n > 4. - Chai Wah Wu, Jun 21 2016

E.g.f.: x*(8 + 54*x + 60*x^2 + 15*x^3)*exp(2*x)/4. - Ilya Gutkovskiy, Jun 21 2016