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A217899 O.g.f.: Sum_{n>=1} (n^2)^(n-1) * exp(-n^2*x) * x^n / n!. 11
1, 1, 6, 65, 1050, 22827, 627396, 20912320, 820784250, 37112163803, 1900842429486, 108823356051137, 6888836057922000, 477898618396288260, 36060660300744309600, 2940812098256837097720, 257780560811305783833450, 24171700822696604400643035, 2414448376056191692970387250 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..250

FORMULA

a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * (k^2)^(n-1).

a(n) = [x^n] x + x^2/Product_{k=1..n} (1-k*x).

a(n) = [x^n] x + x^2*(1+x)^(2*n-3) / Product_{k=1..n-1} (1-k*x).

a(n) = sum(j=0..n-1, binomial(2*n-1,j)*stirling2(2*n-j-1,n)). - Vladimir Kruchinin, Jun 14 2013

a(n) ~ 2^(2*n-5/2) * n^(n-5/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n *(2-c)^(n-2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 20 2014

EXAMPLE

O.g.f.: A(x) = x + x^2 + 6*x^3 + 65*x^4 + 1050*x^5 + 22827*x^6 + 627396*x^7 + ... where A(x) = 1^0*x*exp(-1*x) + 2^2*exp(-2^2*x)*x^2/2! + 3^4*exp(-3^2*x)*x^3/3! + 4^6*exp(-4^2*x)*x^4/4! + 5^8*exp(-5^2*x)*x^5/5! + ... simplifies to a power series in x with integer coefficients.

MATHEMATICA

a[n_] := Sum[ Binomial[2*n - 3, j]*StirlingS2[2*n - j - 3, n-1], {j, 0, n-2}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* Jean-Fran├žois Alcover, Jun 14 2013, after Vladimir Kruchinin *)

PROG

(PARI) {a(n)=polcoeff(sum(m=1, n, (m^2)^(m-1)*x^m*exp(-m^2*x+x*O(x^n))/m!), n)}

(PARI) {a(n)=1/n!*sum(k=1, n, (-1)^(n-k)*binomial(n, k)*(k^2)^(n-1))}

(PARI) {a(n)=polcoeff(x+x^2/prod(k=1, n, 1-k*x +x*O(x^n)), n)}

(PARI) {a(n)=polcoeff(x+x^2*(1+x)^(2*n-3)/prod(k=0, n-1, 1-k*x +x*O(x^n)), n)}

for(n=1, 20, print1(a(n), ", "))

CROSSREFS

Cf. A217900, A217901, A217902, A217903, A217904, A217905.

Sequence in context: A099343 A189509 A243698 * A006959 A121017 A239998

Adjacent sequences:  A217896 A217897 A217898 * A217900 A217901 A217902

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 14 2012

STATUS

approved

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Last modified November 14 12:44 EST 2019. Contains 329116 sequences. (Running on oeis4.)