A226045 G.f.: ( Sum_{n>=0} (2*n+x)^n * x^n / (1 + 2*n*x + x^2)^n )^(1/2).

1, 1, 6, 44, 430, 5162, 72890, 1178978, 21449704, 433116488, 9606793874, 232145293502, 6070097785376, 170763070370848, 5142963967765530, 165115679014587758, 5629558857460143814, 203146937778126705662, 7735490130309647256862

Offset: 0

Author

Paul D. Hanna, May 24 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 44*x^3 + 430*x^4 + 5162*x^5 +...
where the square of the g.f. begins
A(x)^2 = 1 + 2*x + 13*x^2 + 100*x^3 + 984*x^4 + 11712*x^5 + 163200*x^6 +...+ A187746(n)*x^n +...
and equals the series
A(x)^2 = 1 + (2+x)*x/(1+2*x+x^2) + (4+x)^2*x^2/(1+4*x+x^2)^2 + (6+x)^3*x^3/(1+6*x+x^2)^3 + (8+x)^4*x^4/(1+8*x+x^2)^4 + (10+x)^5*x^5/(1+10*x+x^2)^5 +...
		

Programs

PARI

{a(n)=polcoeff( sum(m=0, n, (2*m+x)^m*x^m/(1+2*m*x+x^2 +x*O(x^n))^m)^(1/2), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

Self-convolution yields A187746.

Sum_{k=0..n} a(n)*a(n-k) = (2*n^2 + 2*n + 1) * 2^(n-2) * (n-1)! for n>1.