A094600 G.f. satisfies: a(2*n) equals coefficient of x^n in A(x)^(n+1) and a(2*n+1) equals coefficient of x^(n+1) in A(x)^(n+1), for n>=0, with a(0)=1.

1, 1, 2, 5, 9, 28, 48, 145, 250, 831, 1404, 4664, 7875, 26748, 44960, 154265, 258777, 896644, 1501060, 5239975, 8758640, 30760060, 51350784, 181258264, 302271736, 1071490551, 1785262500, 6351444132, 10574365725, 37738804488, 62788919872

Offset: 0

Author

Paul D. Hanna, May 13 2004

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 28*x^5 + 48*x^6 + 145*x^7 +...
Terms are formed from main and adjacent diagonals in the table of successive self-convolutions of this sequence:
[(1),(1), 2, 5, 9, 28, 48, 145, 250, 831, 1404, 4664,...];
[1,(2),(5), 14, 32, 94, 213, 588, 1343, 3726,...];
[1, 3,(9),(28), 75, 225, 590, 1656, 4287, 11780,...];
[1, 4, 14,(48),(145), 456, 1318, 3864, 10824, 30684,...];
[1, 5, 20, 75,(250),(831), 2590, 7980, 23755, 70155,...];
[1, 6, 27, 110, 399,(1404),(4664), 15102, 47355, 145880,...];
[1, 7, 35, 154, 602, 2240,(7875),(26748), 87892, 282093,...];
[1, 8, 44, 208, 870, 3416, 12648,(44960),(154265), 514920,...];
[1, 9, 54, 273, 1215, 5022, 19512, 72423,(258777),(896644),...]; ...
from which A094601 may be formed from the main diagonal:
[1/1, 2/2, 9/3, 48/4, 250/5, 1404/6, 7875/7, 44960/8, 258777/9,...].
Let G(x) be the g.f. of A094601:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 50*x^4 + 234*x^5 + 1125*x^6 + 5620*x^7 +...
then the logarithm begins:
log(G(x)) = x + 5*x^2/2 + 28*x^3/3 + 145*x^4/4 + 831*x^5/5 + 4664*x^6/6 +...
and is formed from the odd-indexed terms of this sequence.
		

Crossrefs

Cf. A094557, A094601.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n),G);for(i=1,ceil(log(n+1)/log(2)), G=serreverse(x/A)/x; A=subst(G+x*G',x,x^2)+x*subst(G',x,x^2)/subst(G,x,x^2) +x*O(x^n));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

a(2*n) = (n+1)*A094601(n) for n>=0.

Sum_{n>=1} a(2*n-1)*x^n/n = log(G(x)), where G(x) is the g.f. of A094601.

G.f. A(x) satisfies: A(x) = A(x*G(x)), where G(x) is the g.f. of A094601.

G.f.: A(x) = G(x^2) + x*G'(x^2)/G(x^2) + x^2*G'(x^2) where G(x) = (1/x)*Series_Reversion(x/A(x)) is the g.f. of A094601.

Extensions

Entry revised by Paul D. Hanna, Apr 17 2013