A028346 Expansion of 1/((1-x)^4*(1-x^2)^2).

1, 4, 12, 28, 58, 108, 188, 308, 483, 728, 1064, 1512, 2100, 2856, 3816, 5016, 6501, 8316, 10516, 13156, 16302, 20020, 24388, 29484, 35399, 42224, 50064, 59024, 69224, 80784, 93840, 108528, 125001, 143412, 163932, 186732, 212002, 239932, 270732, 304612, 341803

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Equals triangle A152205 as an infinite lower triangular matrix * the triangular numbers: [1, 3, 6, ...]. - Gary W. Adamson, Feb 14 2010

a(n) is the number of partitions of n into four kinds of parts 1 and two kinds of parts 2. - Joerg Arndt, Mar 09 2016

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,10,-4,-4,4,-1).

Crossrefs

Cf. A152205, A001752 (for the similar series 1/((1-x)^4*(1-x^2))).

Programs

Magma

[(n+4)*(2*n^4+32*n^3+172*n^2+352*n+15*(-1)^n+225)/960: n in [0..40]]; // Vincenzo Librandi, Feb 14 2016

Mathematica

CoefficientList[Series[1/((1 - x)^4 (1 - x^2)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2016 *)
LinearRecurrence[{4, -4, -4, 10, -4, -4, 4, -1}, {1, 4, 12, 28, 58, 108, 188, 308}, 100] (* G. C. Greubel, Nov 25 2016 *)

PARI

Vec(1/((1-x)^4*(1-x^2)^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (n+4)*(2*n^4 + 32*n^3 + 172*n^2 + 352*n + 15*(-1)^n + 225)/960. - R. J. Mathar, Apr 01 2010

From Antal Pinter, Jan 08 2016: (Start)

a(n) = C(n + 3, 3) + 2*C(n + 1, 3) + 3*C(n - 1, 3) + 4*C(n - 3, 3) + ...

a(n) = Sum_{i = 1..z} i*C(n + 5 - 2*i,3) where z = (2*n + 3 + (-1)^n)/4.

(End)

a(n) = Sum_{i = 0..n} A002624(i). - Antal Pinter, May 05 2016