A177753 G.f.: A(x) = exp( Sum_{n>=1} (n+1)*A177752(n)*x^n/n - x ).

1, 1, 2, 11, 140, 3102, 102713, 4698780, 283041208, 21704073515, 2064570182438, 238616651727324, 32939304929679337, 5353248306115060288, 1011770777921642230227, 220048666117424880696401

Offset

0,3

Comments

Let G(x) = g.f. of A177752, then A177752 is defined by:

. A177752(n) = [x^n] G(x)^n/(n+1) for n>1.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

G.f. satisfies: 1+x + x*A'(x)/A(x) = d/dx x^2/Series_Reversion(x*A(x)).

a(n) ~ c * (n!)^2 / sqrt(n), where c = 0.500612869985729164508780668394780439... - Vaclav Kotesovec, Oct 18 2017

Example

G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 3102*x^5 +...
Compare the series S(x) = d/dx x^2/Series_Reversion(x*A(x)):
S(x) = 1 + 2*x + 3*x^2 + 28*x^3 + 515*x^4 + 14766*x^5 + 596652*x^6 +...
to the logarithmic derivative:
A'(x)/A(x) = 1 + 3*x + 28*x^2 + 515*x^3 + 14766*x^4 + 596652*x^5 +...
and also to the g.f. G(x) of A177752:
G(x) = 1 + x + x^2 + 7*x^3 + 103*x^4 + 2461*x^5 + 85236*x^6 +...

Prog

(PARI) {a(n)=local(A=1+x+sum(m=2, n-1, a(m)*x^m)); A=(1/x)*serreverse(x^2/intformal(1+x+x*deriv(A)/(A+x*O(x^n)))); if(n<0, 0, if(n<2, 1, polcoeff((n+1)*A, n)))}

Crossrefs

Cf. A177752, A293864.

Sequence in context: A060059 A365357 A295811 * A183609 A113148 A193209

Adjacent sequences: A177750 A177751 A177752 * A177754 A177755 A177756

Keyword

nonn

Author

Paul D. Hanna, May 16 2010

Status

approved