A192935 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n / n!.

1, 1, 4, 39, 592, 12965, 378276, 14062363, 643946920, 35426253465, 2295988778440, 172565368741931, 14847924324645996, 1446814927797156541, 158201328106874927980, 19258822568210913998955, 2592339296719295037808336, 383513887126740027040942577

Offset

0,3

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} (q^n + p)^n * r^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = (1+x) and p = -1, r = 1. - Paul D. Hanna, Jun 21 2019

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..280

Formula

E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(-(1+x)^n) / n!. - Paul D. Hanna, Jun 21 2019

a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A108459(k), where the e.g.f. of A108459 = Sum_{n>=0} (exp(n*x)-1)^n/n! (see Vladeta Jovovic's formula in A122400).

Example

E.g.f: A(x) = 1 + x + 4*x^2/2! + 39*x^3/3! + 592*x^4/4! + 12965*x^5/5! + 378276*x^6/6! + 14062363*x^7/7! + 643946920*x^8/8! + 35426253465*x^9/9! + 2295988778440*x^10/10! +...
such that
A(x) = 1 + ((1+x) - 1) + ((1+x)^2 - 1)^2/2! + ((1+x)^3 - 1)^3/3! + ((1+x)^4 - 1)^4/4! + ((1+x)^5 - 1)^5/5! + ((1+x)^6 - 1)^6/6! + ((1+x)^7 - 1)^7/7! + ...
also
A(x) = 1 + (1+x)*exp(-(1+x)) + (1+x)^4*exp(-(1+x)^2)/2! + (1+x)^9*exp(-(1+x)^3)/3! + (1+x)^16*exp(-(1+x)^4)/4! + (1+x)^25*exp(-(1+x)^5)/5! + (1+x)^36*exp(-(1+x)^6)/6! + (1+x)^49*exp(-(1+x)^7)/7! + ...
RELATED SERIES.
Expansion of ((1+x)^n-1)^n suggests that the e.g.f. is related to LambertW(x):
((1+x)^2-1)^2 = 4*x^2 + 4*x^3 + x^4;
((1+x)^3-1)^3 = 27*x^3 + 81*x^4 + 108*x^5 + 81*x^6 + 36*x^7 + 9*x^8 + x^9;
((1+x)^4-1)^4 = 256*x^4 + 1536*x^5 + 4480*x^6 + 8320*x^7 + 10896*x^8 +...

Mathematica

With[{m = 20}, CoefficientList[Series[Sum[If[n==0, 1, ((1+x)^n -1)^n/n!], {n, 0, m+2}], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 06 2019 *)

Prog

(PARI) {a(n)=n!*polcoeff(sum(m=0, n, ((1+x+x*O(x^n))^m-1)^m/m!), n)}
for(n=0, 30, print1(a(n)*n!, ", "))
(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0, n, Stirling1(n, k)*k!*polcoeff(sum(m=0, k, (exp(m*x+x*O(x^n))-1)^m/m!), k))}
for(n=0, 30, print1(a(n)*n!, ", "))
(Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[((1+x)^n -1)^n/Factorial(n): n in [0..m+2]]) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 06 2019
(Sage) m = 20; T = taylor(sum(((1+x)^k-1)^k/factorial(k) for k in range(m+2)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 06 2019

Crossrefs

Cf. A192985, A108459, A122400.

Sequence in context: A300188 A177775 A364981 * A365010 A024055 A361544

Adjacent sequences: A192932 A192933 A192934 * A192936 A192937 A192938

Keyword

nonn

Author

Paul D. Hanna, Jul 13 2011

Status

approved