A220179 E.g.f.: Sum_{n>=1} (1 - exp(-n^2*x))^n / n.

1, 15, 1267, 316275, 174397531, 179770837155, 310789895286907, 834906367019076675, 3293344593080631993211, 18259284528276047000517795, 137429981152689382429349060347, 1365009985652048448232840864764675, 17475885712645599218827214639383437691

Offset

1,2

Comments

Compare to the trivial identity: x = Sum_{n>=1} (1 - exp(-x))^n/n.

Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..162

Formula

O.g.f.: Sum_{n>=1} n^(2*n-1) * n! * x^n / Product_{k=1..n} (1 - n^2*k*x). - Paul D. Hanna, Jan 05 2013

a(n) = Sum_{k=1..n} (-1)^(n-k) * k^(2*n-1) * k! * Stirling2(n,k). - Paul D. Hanna, Jan 05 2013

a(n) ~ c * d^n * (n!)^3 / n^2, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.175744118254830086361220160145768507562830495967... . - Vaclav Kotesovec, May 08 2014

Example

E.g.f.: A(x) = x + 15*x^2/2! + 1267*x^3/3! + 316275*x^4/4! + 174397531*x^5/5! +...
where
A(x) = (1-exp(-x)) + (1-exp(-4*x))^2/2 + (1-exp(-9*x))^3/3 + (1-exp(-16*x))^4/4 + (1-exp(-25*x))^5/5 +...

Mathematica

Table[Sum[(-1)^(n-k) * k^(2*n-1) * k! * StirlingS2[n, k], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, May 08 2014 *)

Prog

(PARI) a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k^2*x+x*O(x^n)))^k/k), n)
for(n=1, 20, print1(a(n), ", "))
(PARI) a(n)=polcoeff(sum(m=1, n, m^(2*m-1)*m!*x^m/prod(k=1, m, 1+m^2*k*x+x*O(x^n))), n) \\ Paul D. Hanna, Jan 05 2013
for(n=1, 20, print1(a(n), ", "))
(PARI) {a(n)=sum(k=1, n, (-1)^(n-k)*k^(2*n-1)*k!*stirling(n, k, 2))}
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013

Crossrefs

Cf. A092552, A187755, A220181, A242228.

Sequence in context: A135810 A273967 A273963 * A373858 A166330 A300618

Adjacent sequences: A220176 A220177 A220178 * A220180 A220181 A220182

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2012

Status

approved