A189427 Expansion of (x^2)/((1-x)*(1-2*x-x^2+x^3)^2).

0, 0, 1, 5, 19, 61, 180, 502, 1349, 3529, 9050, 22854, 57014, 140832, 345036, 839530, 2030757, 4887423, 11710757, 27951471, 66486128, 157661282, 372840407, 879510801, 2070045268, 4862121660, 11398688956, 26676792832, 62333380456, 145434747140

Offset: 0

Author

L. Edson Jeffery, Apr 22 2011

Keywords

• nonn
• easy

Comments

Second of a series of sequences of partial sums of (nonzero) diagonals of triangle A188106 whose diagonals correspond to successive convolutions of A006054 with itself, where the first such sequence of partial sums is given by A077850. For n=1,2,..., this series of sequences is generated by successive series expansion of 1/((1-x)*(1-2*x-x^2+x^3)^n), for which A077850 corresponds to n=1 and A189427 corresponds to n=2.

a(n)=Sum_{k=0..n} A189426(k), where A189426={0,0,1,4,14,42,119,322,...} is the convolution of A006054={0,0,1,2,5,11,25,56,126,...} with itself. Also, a(n+2)=Sum_{k=0..n} A188106{n+k+1,k}, n=0,1,2,....

Crossrefs

Cf. A006054, A077850, A188106, A189426.

Programs

PARI

Vec((x^2)/((1-x)*(1-2*x-x^2+x^3)^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

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

a(n)=5*a(n-1)-6*a(n-2)-4*a(n-3)+9*a(n-4)-a(n-5)-3*a(n-6)+a(n-7), n>=7, a{m}={0,0,1,5,19,61,180}, m=0..6.