A195736 E.g.f.: x = Sum_{n>=1} a(n)*x^n/n! * exp(-n^2*x).

1, 2, 21, 568, 29705, 2573136, 335201293, 61480323584, 15135660248913, 4823681315219200, 1934425407465004421, 954153609788873382912, 568125617688093236137561, 402006917909739659429470208, 333597313002114320208678928125

Offset

1,2

Comments

Compare e.g.f. to: x = Sum_{n>=1} n^(n-1)*x^n/n! * exp(-n*x), which generates coefficients for the series reversion of x*exp(-x).

Links

Table of n, a(n) for n=1..15.

Formula

G.f.: x = Sum_{n>=1} a(n)*x^n/(n*(1 + n^2*x)^n).

a(n) = n*A082157(n+1).

Example

x = x*exp(-x) + 2*x^2/2!*exp(-4*x) + 21*x^3/3!*exp(-9*x) + 568*x^4/4!*exp(-16*x) + 29705*x^5/5!*exp(-25*x) +...+ a(n)*x^n/n!*exp(-n^2*x) +...
The coefficients a(n) also satisfy:
x = x/(1+x) + 2*x^2/(2*(1+4*x)^2) + 21*x^3/(3*(1+9*x)^3) + 568*x^4/(4*(1+16*x)^4) + 29705*x^5/(5*(1+25*x)^5) +...+ a(n)*x^n/(n*(1+n^2*x)^n) +...

Prog

(PARI) {a(n)=if(n<1, 0, n!*polcoeff(x-sum(m=1, n-1, a(m)*x^m/m!*exp(-m^2*x+x*O(x^n))), n))}
(PARI) {a(n)=if(n<1, 0, n*polcoeff(x-sum(m=1, n-1, a(m)*x^m/(m*(1+m^2*x+x*O(x^n))^m)), n))}

Crossrefs

Cf. A195737, A082157.

Sequence in context: A356481 A171107 A218768 * A320653 A302686 A078602

Adjacent sequences: A195733 A195734 A195735 * A195737 A195738 A195739

Keyword

nonn

Author

Paul D. Hanna, Sep 30 2011

Status

approved