A232601 a(n) = Sum_{k=0..n} k^p*q^k for p = 2 and q = -2.

0, -2, 14, -58, 198, -602, 1702, -4570, 11814, -29658, 72742, -175066, 414758, -969690, 2241574, -5131226, 11645990, -26233818, 58700838, -130567130, 288863270, -635980762, 1394062374, -3043511258, 6620165158

Offset

0,2

Links

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Stanislav Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.

Index entries for linear recurrences with constant coefficients, signature (-5,-6,4,8).

Formula

a(n) = 2*((-2)^n * (9*n^2 + 6*n - 1) + 1)/27.

G.f.: 2*x*(-1 + 2*x) / ((1-x)*(1+2*x)^3). - R. J. Mathar, Nov 23 2014

E.g.f.: (2/27)*(exp(x) - (1 +30*x -36*x^2)*exp(-2*x)). - G. C. Greubel, Mar 31 2021

a(n) = - 5*a(n-1) - 6*a(n-2) + 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

Example

a(3) = 0^2*2^0 - 1^2*2^1 + 2^2*2^2 - 3^2*2^3 = -58.

Maple

A232601:= n-> 2*(1 - (-2)^n*(1-6*n-9*n^2))/27; seq(A232601(n), n=0..30); # G. C. Greubel, Mar 31 2021

Mathematica

LinearRecurrence[{-5, -6, 4, 8}, {0, -2, 14, -58}, 30] (* Harvey P. Dale, Aug 20 2015 *)

Prog

(PARI) S2M2(n)=((-1)^n*2^(n+1)*(9*n^2+6*n-1)+2)/27;
v = vector(10001); for(k=1, #v, v[k]=S2M2(k-1))
(Magma) [2*(1 - (-2)^n*(1-6*n-9*n^2))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021
(Sage) [2*(1 - (-2)^n*(1-6*n-9*n^2))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

Crossrefs

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

Sequence in context: A115027 A114146 A096367 * A285153 A232370 A174704

Adjacent sequences: A232598 A232599 A232600 * A232602 A232603 A232604

Keyword

sign,easy

Author

Stanislav Sykora, Nov 27 2013

Status

approved