A276370 G.f. A(x) satisfies: A( x - A(x) ) = x^2.

1, 2, 9, 56, 420, 3572, 33328, 334354, 3559310, 39838760, 465743720, 5658983108, 71191948512, 924554859776, 12365546196641, 169995491295312, 2398380272232272, 34680290150700800, 513390937937217088, 7773229533145403728, 120277760289804227632, 1900583166564027019136, 30649888151334972466392, 504153517331248726221392, 8454018409655883681321232, 144451967918022160558965408

Offset: 2

Author

Paul D. Hanna, Sep 09 2016

Keywords

• nonn

Comments

An unsigned version of A138740 (apart from initial term).

Self-convolution of A213591.

Examples

			G.f.: A(x) = x^2 + 2*x^3 + 9*x^4 + 56*x^5 + 420*x^6 + 3572*x^7 + 33328*x^8 + 334354*x^9 + 3559310*x^10 + 39838760*x^11 + 465743720*x^12 +...
such that A( x - A(x) ) = x^2.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 2..302

Crossrefs

Cf. A138740, A213591, A212411, A213628.

Programs

PARI

{a(n) = local(A=x^2); for(i=1, n, A = serreverse(x - A +x*O(x^n))^2); polcoeff(A, n)}
for(n=2,30,print1(a(n),", "))

PARI

{Dx(n, F) = local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = local(A=x^2 +x*O(x^n)); for(i=1, n, A = (x + sum(m=1, n, Dx(m-1, A^m)/m!) +x*O(x^n))^2); polcoeff(A, n)}
for(n=2,30,print1(a(n),", "))

Formula

G.f. satisfies:

(1) A(x) = ( x + A( sqrt(A(x)) ) )^2.

(2) A(x) = ( x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! )^2.

(3) A(x) = x^2 * exp( 2*Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n/x / n! ).

(4) A(x) = x^2 * G( A(x)/x )^2 where G(x) = 1 + x*G( 1 - 1/G(x) )^2 is the g.f. of A212411.

(5) A(x) = x^2 * F( sqrt(A(x)) )^2 where F(x) = 1 - (x^2/F(x)) / F( x^2/F(x) ) is the g.f. of A213628.