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A217911 O.g.f.: Sum_{n>=0} n^n * (3*n+1)^(n-1) * exp(-n*(3*n+1)*x) * x^n / n!. 3
1, 1, 10, 262, 11296, 684172, 53598952, 5162269744, 590636585728, 78321222303184, 11815503098606560, 1998732510370890208, 374763163567227915520, 77151431783218955979520, 17301697176590720940003328, 4198491769695976346962419712, 1096165878182404669364316147712 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} (3*n+1)^(n-1) * exp(-(3*n+1)*x) * x^n/n!.

LINKS

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

FORMULA

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (3*k+1)^(n-1).

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(3*k+1)^(k-1)*x^k / (1 + k*(3*k+1)*x)^(k+1).

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

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

a(n) ~ 2^(2*n-4/3) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/6)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

EXAMPLE

O.g.f.: A(x) = 1 + x + 10*x^2 + 262*x^3 + 11296*x^4 + 684172*x^5 +...

where

A(x) = 1 + 1^1*4^0*x*exp(-1*4*x) + 2^2*7^1*exp(-2*7*x)*x^2/2! + 3^3*10^2*exp(-3*10*x)*x^3/3! + 4^4*13^3*exp(-4*13*x)*x^4/4! + 5^5*16^4*exp(-5*16*x)*x^5/5! +...

simplifies to a power series in x with integer coefficients.

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n-1, j]*3^j*StirlingS2[n+j, n], {j, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2014 *)

PROG

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

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

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

(PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(3*k+1)^(n-1))}

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

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

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

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

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

CROSSREFS

Cf. A217900, A217910, A217912.

Sequence in context: A183406 A084999 A268730 * A054593 A290041 A160481

Adjacent sequences:  A217908 A217909 A217910 * A217912 A217913 A217914

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 14 2012

STATUS

approved

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Last modified June 26 00:10 EDT 2019. Contains 324367 sequences. (Running on oeis4.)