A340939 E.g.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*A(x)^(2*n)*exp(x*A(x)^n)/n! and Q(x) = Sum_{n>=0} x^n*A(x)^n*exp(x*A(x)^(n+1))/n!.

1, 1, 4, 42, 648, 13620, 362520, 11696160, 443748480, 19362566160, 955384416000, 52602602199840, 3197371241877120, 212668103188981440, 15364762939152506880, 1198224396270059558400, 100322035878465399398400, 8975632678570151712518400, 854594260998625751469465600

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f. A(x) satisfies: A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (n+1) * x^n * A(x)^(2*n) * exp(x*A(x)^n) / n!,

Q(x) = Sum_{n>=0} x^n * A(x)^n * exp(x*A(x)^(n+1)) / n!.

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 648*x^4/4! + 13620*x^5/5! + 362520*x^6/6! + 11696160*x^7/7! + 443748480*x^8/8! + 19362566160*x^9/9! + ...
such that A(x) = P(x)/Q(x) where
P(x) = exp(x) + 2*x*A(x)^2*exp(x*A(x)) + 3*x^2*A(x)^4*exp(x*A(x)^2)/2! + 4*x^3*A(x)^6*exp(x*A(x)^3)/3! + 5*x^4*A(x)^8*exp(x*A(x)^4)/4! + ...
Q(x) = exp(x*A(x)) + x*A(x)*exp(x*A(x)^2) + x^2*A(x)^2*exp(x*A(x)^3)/2! + x^3*A(x)^3*exp(x*A(x)^4)/3! + x^4*A(x)^4*exp(x*A(x)^5)/4! + ...
explicitly,
P(x) = 1 + 3*x + 16*x^2/2! + 152*x^3/3! + 2256*x^4/4! + 46172*x^5/5! + 1207456*x^6/6! + 38466192*x^7/7! + 1445453344*x^8/8! + 62597927312*x^9/9! + ...
Q(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 832*x^4/4! + 16072*x^5/5! + 405304*x^6/6! + 12590216*x^7/7! + 464416544*x^8/8! + 19828679264*x^9/9! + ...

Prog

(PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1);
for(i=0, n,
P = sum(m=0, n, (m+1)*x^m*A^(2*m)/m! * exp(x*A^m + x*O(x^n)) );
Q = sum(m=0, n, x^m*A^m/m! * exp(x*A^(m+1) + x*O(x^n)) );
A = P/Q); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A340942, A340938.

Sequence in context: A153854 A216080 A137645 * A355410 A136045 A192949

Adjacent sequences: A340936 A340937 A340938 * A340940 A340941 A340942

Keyword

nonn

Author

Paul D. Hanna, Feb 08 2021

Status

approved