A191505 G.f.: 1 = Sum_{n>=0} a(n)*exp(-n!*x)*x^n/n!.

1, 1, 1, 4, 75, 7636, 4866965, 22256484426, 827473662052359, 280073424855627741304, 956136927041635596586248969, 36146438316110599447497305174316790, 16486165931975571004114967909531846539984907

Offset

0,4

Links

Table of n, a(n) for n=0..12.

Formula

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

1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + n!*x)^(n+m) for m>=1.

log(1+x) = Sum_{n>=1} a(n)*x^n/(1 + n!*x)^n/n.

a(n) = Sum_{k>=0..n-1} -(-1)^(n-k)*C(n,k)*k!^(n-k)*a(k) for n>0 with a(0)=1.

Example

1 = exp(-x) + exp(-x)*x + exp(-2!*x)*x^2/2! + 4*exp(-3!*x)*x^3/3! + 75*exp(-4!*x)*x^4/4! +...
1 = 1/(1+x) + x/(1+x)^2 + x^2/(1+2!*x)^3 + 4*x^3/(1+3!*x)^4 + 75*x^4/(1+4!*x)^5 +...
1 = 1/(1+x)^2 + 1*2*x/(1+x)^3 + 1*3*x^2/(1+2!*x)^4 + 4*4*x^3/(1+3!*x)^5 + 75*5*x^4/(1+4!*x)^6 +...
1 = 1/(1+x)^3 + 1*3*x/(1+x)^4 + 1*6*x^2/(1+2!*x)^5 + 4*10*x^3/(1+3!*x)^6 + 75*15*x^4/(1+4!*x)^7 +...
log(1+x) = x/(1+x) + x^2/(1+2!*x)^2/2 + 4*x^3/(1+3!*x)^3/3 + 75*x^4/(1+4!*x)^4/4 + 7636*x^5/(1+5!*x)^5/5 + 4866965*x^6/(1+6!*x)^6/6 +...

Prog

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

Crossrefs

Sequence in context: A006236 A374884 A120248 * A100323 A262073 A067921

Adjacent sequences: A191502 A191503 A191504 * A191506 A191507 A191508

Keyword

nonn

Author

Paul D. Hanna, Jun 04 2011

Status

approved