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

0,4

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,....

Links

Table of n, a(n) for n=0..29.

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.

Prog

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

Crossrefs

Cf. A006054, A077850, A188106, A189426.

Sequence in context: A036637 A036644 A000342 * A355492 A212339 A072111

Adjacent sequences: A189424 A189425 A189426 * A189428 A189429 A189430

Keyword

nonn,easy

Author

L. Edson Jeffery, Apr 22 2011

Status

approved