A219119 Expansion of e.g.f.: Sum_{n>=0} log(1 + x/(1-x)^n)^n / n!.

1, 1, 2, 12, 96, 1000, 13500, 221718, 4301808, 97747200, 2555001360, 75526842600, 2503943418240, 92407030642056, 3759862792921872, 167429488088545200, 8120958429706093440, 426777425467443381120, 24161214872571127574400, 1467122583066982481802816

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..300

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! + ...

Mathematica

m:= 50;
f[x_, m_]:= Sum[Product[(1/(1-x)^n -j), {j, 0, n-1}]*x^n/n!, {n, 0, m+1}];
CoefficientList[Series[f[x, m], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 02 2023 *)

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), ", "))
(Magma)
m:=50;
f:= func< x | 1 + (&+[(&*[1/(1-x)^n -j: j in [0..n-1]])*x^n/Factorial(n) : n in [1..m+2]]) >;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Feb 02 2023
(SageMath)
m=50
def f(x): return sum(product(1/(1-x)^n - k for k in range(n))*x^n/factorial(n) for n in range(m+2))
def A219119_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A219119_list(m) # G. C. Greubel, Feb 02 2023

Crossrefs

Cf. A216839.

Sequence in context: A014297 A193425 A206855 * A052611 A340938 A059864

Adjacent sequences: A219116 A219117 A219118 * A219120 A219121 A219122

Keyword

sign

Author

Paul D. Hanna, Nov 13 2012

Status

approved