A291848 G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + (2*k+1)*x + x^2).

1, 1, 2, 6, 15, 47, 150, 522, 1903, 7319, 29396, 122988, 534141, 2400061, 11136516, 53220492, 261576725, 1319629445, 6825232486, 36137198722, 195664517227, 1082169511883, 6108213101658, 35153836421302, 206126910439763, 1230477025952427, 7473067121404104, 46146114390128888, 289554642297817561, 1845220293901278041, 11936266843924805064

Offset: 0

Author

Paul D. Hanna, Sep 03 2017

Keywords

• nonn

Comments

Antidiagonal sums of irregular triangle A291845, which has row sums equal to the odd double factorials A001147.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 47*x^5 + 150*x^6 + 522*x^7 + 1903*x^8 + 7319*x^9 + 29396*x^10 + 122988*x^11 + 534141*x^12 +...
which equals the series:
A(x) = 1 + x*(1+x+x^2) + x^2*(1+x+x^2)*(1+3*x+x^2) + x^3*(1+x+x^2)*(1+3*x+x^2)*(1+5*x+x^2) + x^4*(1+x+x^2)*(1+3*x+x^2)*(1+5*x+x^2)*(1+7*x+x^2) +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A291845, A201951.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Sum[2^n*x^(2*n)*Pochhammer[(1 + x + x^2)/(2*x), n], {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2017 *)

PARI

{a(n)=sum(k=0, n, polcoeff(prod(j=0, n-k-1, 1+(2*j+1)*x+x^2), k))}
for(n=0,30,print1(a(n),", "))

PARI

{a(n)=polcoeff(sum(m=0, n, x^m*prod(j=0, m-1, 1+(2*j+1)*x+x^2))+x*O(x^n), n)}
for(n=0,25,print1(a(n),", "))