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A232604 2^n*sum( k=0..n, k^p*q^k ), where p=3, q=-1/2. 8
0, -1, 6, -15, 34, -57, 102, -139, 234, -261, 478, -375, 978, -241, 2262, 1149, 6394, 7875, 21582, 36305, 80610, 151959, 314566, 616965, 1247754, 2479883, 4977342, 9935001, 19891954, 39759519, 79546038, 159062285 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The factor 2^n (i.e. |1/q|^n) is present to make the values integers.

See also A232600 and references therein for integer values of q.

The same values with different signs are produced by a(n) = n^3 - 2*a(n). The signs are all positive until n = 15, with negative signs on values for all subsequent odd indices. - Richard R. Forberg, Feb 17 2014.

LINKS

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

S. 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 (-2,2,8,7,2).

FORMULA

a(n) = (2^(n+1)+(-1)^n*(9*n^3+18*n^2+6*n-2))/27.

G.f.: x*(1-4*x+x^2) / ( (2*x-1)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014

EXAMPLE

a(3) = 2^3 * (0^3/2^0 - 1^3/2^1 + 2^3/2^2 - 3^3/2^3) = 0-4+16-27 = -15.

PROG

(PARI) a(n)=(2^(n+1)+(-1)^n*(9*n^3+18*n^2+6*n-2))/27;

CROSSREFS

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232603 (p=2,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

Sequence in context: A051410 A083052 A265395 * A120849 A030661 A245630

Adjacent sequences:  A232601 A232602 A232603 * A232605 A232606 A232607

KEYWORD

sign,easy

AUTHOR

Stanislav Sykora, Nov 27 2013

STATUS

approved

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Last modified May 26 03:36 EDT 2017. Contains 287073 sequences.