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A219119 E.g.f.: Sum_{n>=0} log(1 + x/(1-x)^n)^n / n!. 0
1, 1, 2, 12, 96, 1000, 13500, 221718, 4301808, 97747200, 2555001360, 75526842600, 2503943418240, 92407030642056, 3759862792921872, 167429488088545200, 8120958429706093440, 426777425467443381120, 24161214872571127574400, 1467122583066982481802816 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

FORMULA

E.g.f.: Sum_{n>=0} binomial(1/(1-x)^n, n) * x^n.

E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1/(1-x)^n - k).

E.g.f.: Sum_{n>=0} x^n/n! * Sum_{k=0..n} Stirling1(n,k) / (1-x)^(n*k).

EXAMPLE

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 96*x^4/4! + 1000*x^5/5! +...

where the g.f. satisfies the identities:

A(x) = 1 + log(1+x/(1-x)) + log(1+x/(1-x)^2)^2/2! + log(1+x/(1-x)^3)^3/3! + log(1+x/(1-x)^4)^4/4! + log(1+x/(1-5*x)^5)^5/5! +...

A(x) = 1 + binomial(1/(1-x),1)*x + binomial(1/(1-x)^2,2)*x^2 + binomial(1/(1-x)^3,3)*x^3 + binomial(1/(1-x)^4,4)*x^4 + binomial(1/(1-x)^5,5)*x^5 +...

A(x) = 1 + x/(1-x) + x^2/(1-x)^4*(1-(1-x)^2)/2! + x^3/(1-x)^9*(1-(1-x)^3)*(1-2*(1-x)^3)/3! + x^4/(1-x)^16*(1-(1-x)^4)*(1-2*(1-x)^4)*(1-3*(1-x)^4)/4! +...

PROG

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

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

for(n=0, 30, print1(a(n), ", "))

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

(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

{a(n)=local(A=1+x); A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)/(1-x+x*O(x^n))^(m*k))*x^m/m!); n!*polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A216839.

Sequence in context: A014297 A193425 A206855 * A052611 A059864 A095338

Adjacent sequences:  A219116 A219117 A219118 * A219120 A219121 A219122

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 13 2012

STATUS

approved

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Last modified October 16 20:55 EDT 2019. Contains 328103 sequences. (Running on oeis4.)