A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A003701(n)*x^n.

1, 1, 1, 0, 3, 0, 38, 0, 947, 0, 37394, 0, 2120190, 0, 162980012, 0, 16330173251, 0, 2070201641498, 0, 324240251016266, 0, 61525045423103316, 0, 13913915097436287598, 0, 3698477457114061621492, 0

Offset: 0

Author

Paul D. Hanna, Feb 07 2010

Keywords

• nonn

Comments

The e.g.f. of A003701 is exp(x)/cos(x) = Sum_{n>=0} A003701(n)*x^n/n!.

Compare to A157308 and A157310.

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^4 + 38*x^6 + 947*x^8 + 37394*x^10 +...
where G(x) = A(x*G(x)) is the o.g.f. of A003701:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 +...
while the e.g.f. of A003701 is given by:
exp(x)/cos(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 12*x^4/4! + 36*x^5/5! +...
		

Crossrefs

Cf. A003701, A158119, A157308, A157310.

Programs

PARI

{a(n)=local(X=x+x*O(x^n),G=sum(m=0,n,m!*polcoeff(exp(X)/cos(X),m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}

Formula

a(n) = |A157308(n)| = |A157310(n)| for n>=0.

a(2n) = A158119(n) for n>=0; a(2n-1) = 0 for n>=2, with a(1)=1.

G.f. A = A(x) satisfies: A(x) = 1/(1-x/A - (x/A)^2/(1-x/A - 2^2*(x/A)^2/(1-x/A - 3^2*(x/A)^2/(1-x/A - 4^2*(x/A)^2/(1-x/A - 5^2*(x/A)^2/(1-x/A -...)))))), a recursive continued fraction. [From Paul D. Hanna, Jan 05 2012]