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A222526 O.g.f.: Sum_{n>=0} (n^6)^n * exp(-n^6*x) * x^n / n!. 5
1, 1, 2047, 64439010, 11681056634501, 7713000216608565075, 14204422416132896951197888, 61232072982330045410678351728440, 545827051514425992551826008968173372261, 9173647538352903119028122246836507680995590680 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
a(n) = Stirling2(6*n, n).
a(n) = [x^(6*n)] (6*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(5*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^6)^k*x^k / (1 + k^6*x)^(k+1).
a(n) ~ n^(5*n) * 6^(6*n) / (sqrt(2*Pi*(1-c)*n) * exp(5*n) * (6-c)^(5*n) * c^n), where c = -LambertW(-6*exp(-6)). - Vaclav Kotesovec, May 11 2014
EXAMPLE
O.g.f.: A(x) = 1 + x + 2047*x^2 + 64439010*x^3 + 11681056634501*x^4 +...+ Stirling2(6*n, n)*x^n +...
where
A(x) = 1 + 1^6*x*exp(-1^6*x) + 2^12*exp(-2^6*x)*x^2/2! + 3^18*exp(-3^6*x)*x^3/3! + 4^24*exp(-4^6*x)*x^4/4! + 5^30*exp(-5^6*x)*x^5/5! +...
is a power series in x with integer coefficients.
MATHEMATICA
Table[StirlingS2[6*n, n], {n, 0, 20}] (* Vaclav Kotesovec, May 11 2014 *)
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n, (k^6)^k*exp(-k^6*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^6)^k*x^k/(1+k^6*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(5*n))), 5*n)}
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(6*n, n)}
for(n=0, 12, print1(a(n), ", "))
CROSSREFS
Sequence in context: A236373 A289476 A353409 * A035892 A069272 A234881
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 23 2013
STATUS
approved

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Last modified July 19 15:48 EDT 2024. Contains 374410 sequences. (Running on oeis4.)