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A229261 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / Product_{k=1..n} (1 - n^2*k*x). 7
1, 1, 17, 922, 106695, 21742971, 6977367418, 3273755821827, 2129976884025085, 1846718792259030760, 2068516760060790309349, 2919795339100534415091143, 5088912154987483773753872912, 10766599670032172748225017763021, 27254500086981764567988714050736205 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..14.

FORMULA

a(n) = Sum_{k=0..n} k^(2*n) * Stirling2(n, k).

E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n!.

EXAMPLE

O.g.f.: A(x) = 1 + x + 17*x^2 + 922*x^3 + 106695*x^4 + 21742971*x^5 +...

where

A(x) = 1 + x/(1-x) + 2^4*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3^6*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4^8*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...

Exponential Generating Function.

E.g.f.: E(x) = 1 + x + 17*x^2/2! + 922*x^3/3! + 106695*x^4/4! +...

where

E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2! + (exp(9*x)-1)^3/3! + (exp(16*x)-1)^4/4! + (exp(25*x)-1)^5/5! + (exp(36*x)-1)^6/6! +...

MATHEMATICA

Flatten[{1, Table[Sum[k^(2*n) * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 08 2014 *)

PROG

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

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=n!*polcoeff(sum(m=0, n, (exp(m^2*x+x*O(x^n))-1)^m/m!), n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}

{a(n)=sum(k=0, n, k^(2*n) * Stirling2(n, k))}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A229257, A229258, A229259, A229260, A229233, A229234, A220181, A122399.

Sequence in context: A218660 A086265 A156138 * A196873 A262386 A298302

Adjacent sequences:  A229258 A229259 A229260 * A229262 A229263 A229264

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 17 2013

STATUS

approved

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Last modified March 19 00:15 EDT 2019. Contains 321306 sequences. (Running on oeis4.)