A089658 a(n) = S1(n,1), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

0, 2, 11, 42, 136, 400, 1104, 2912, 7424, 18432, 44800, 107008, 251904, 585728, 1347584, 3072000, 6946816, 15597568, 34799616, 77201408, 170393600, 374341632, 818937856, 1784676352, 3875536896, 8388608000, 18102616064, 38956695552, 83617644544, 179046449152

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 (6,-12,8).

Formula

a(n) = n*(5 + 3*n) * 2^(n-3). (See Wang and Zhang p. 333.)

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

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 2.

G.f.: x*(2 - x)/(1 - 2*x)^3. (End)

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

a(n) = 2*A001788(n) - A001788(n-1). - R. J. Mathar, Jul 22 2021

Mathematica

LinearRecurrence[{6, -12, 8}, {0, 2, 11}, 40] (* Vincenzo Librandi, Jun 22 2016 *)

Prog

(Magma) I:=[0, 2, 11]; [n le 3 select I[n] else 6*Self(n-1)-12*Self(n-2)+8*Self(n-3): n in [1..41]]; // Vincenzo Librandi, Jun 22 2016
(SageMath) [n*(5+3*n)*2^(n-3) for n in (0..40)] # G. C. Greubel, May 24 2022

Crossrefs

Sequences of S1(n, t): A001792 (t=0), this sequence (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

Sequence in context: A107020 A160945 A079808 * A219100 A140322 A027247

Adjacent sequences: A089655 A089656 A089657 * A089659 A089660 A089661

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 04 2004

Status

approved