A292551 Expansion of x*(1 - 2*x + x^2 + 7*x^3 - x^4)/((1 - x)^4*(1 + x)^3).

0, 1, -1, 3, 4, 12, 21, 34, 56, 75, 115, 141, 204, 238, 329, 372, 496, 549, 711, 775, 980, 1056, 1309, 1398, 1704, 1807, 2171, 2289, 2716, 2850, 3345, 3496, 4064, 4233, 4879, 5067, 5796, 6004, 6821, 7050, 7960, 8211, 9219, 9493, 10604, 10902, 12121, 12444, 13776, 14125, 15575

Offset

0,4

Comments

The n-th generalized n-gonal number (for n >= 5).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Formula

G.f.: x*(1 - 2*x + x^2 + 7*x^3 - x^4)/((1 - x)^4*(1 + x)^3).

a(n) = [x^n] x*(1 + (n - 4)*x + x^2)/((1 - x)^3*(1 + x)^2).

E.g.f.: (1/16)*((-6 + 9*x + 2*x^2)*exp(-x) + (6 - 5*x + 4*x^2 + 2*x^3)*exp(x)).

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).

a(n) = (2*n^3 - 2*n^2 + (-1)^n*(2*n^2 - 11*n - 6) - 5*n + 6)/16.

Maple

a:= n-> (m-> m*((n-2)*m-(n-4))/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 29 2018

Mathematica

CoefficientList[Series[x (1 - 2 x + x^2 + 7 x^3 - x^4)/((1 - x)^4 (1 + x)^3), {x, 0, 50}], x]
Table[SeriesCoefficient[x (1 + (n - 4) x + x^2)/((1 - x)^3 (1 + x)^2), {x, 0, n}], {n, 0, 50}]
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 1, -1, 3, 4, 12, 21}, 51]
Table[(2 n^3 - 2 n^2 + (-1)^n (2 n^2 - 11 n - 6) - 5 n + 6)/16, {n, 0, 50}]

Prog

(PARI) x='x+O('x^99); concat(0, Vec(x*(1-2*x+x^2+7*x^3-x^4)/((1-x)^4*(1+x)^3))) \\ Altug Alkan, Sep 18 2017
(GAP) List([0..50], n->(2*n^3-2*n^2+(-1)^n*(2*n^2-11*n-6)-5*n+6)/16); # Muniru A Asiru, Aug 08 2018

Crossrefs

Cf. A006484, A060354, A100119, A195152.

Main diagonal of A303301.

Sequence in context: A282220 A281892 A101727 * A227429 A075220 A075221

Adjacent sequences: A292548 A292549 A292550 * A292552 A292553 A292554

Keyword

easy,sign

Author

Ilya Gutkovskiy, Sep 18 2017

Status

approved