A221214 O.g.f.: Sum_{n>=0} (3*n+1)^(3*n+1) * exp(-(3*n+1)^3*x) * x^n / n!.
1, 255, 395388, 1525953330, 10977340509135, 126827739333023274, 2148335345336441463090, 50163717301669569182864400, 1544377393328765493716910877185, 60615459491155396034172113103266025, 2954227738557038665136475801709196246304
Offset: 0
Keywords
Examples
O.g.f.: A(x) = 1 + 255*x + 395388*x^2 + 1525953330*x^3 + 10977340509135*x^4 +... where A(x) = exp(-x) + 4^4*x*exp(-4^3*x) + 7^7*exp(-7^3*x)*x^2/2! + 10^10*exp(-10^3*x)*x^3/3! + 13^13*exp(-13^3*x)*x^4/4! + 16^16*exp(-16^3*x)*x^5/5! +... is a power series in x with integer coefficients.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..159
Programs
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Mathematica
Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(3*k+1)^(3*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *) Table[Sum[Binomial[3*n+1,n+k]*3^(n+k)*StirlingS2[n+k,n],{k,0,2*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *) -
PARI
{a(n)=polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*exp(-(3*k+1)^3*x +x*O(x^n))*x^k/k!), n)} for(n=0, 20, print1(a(n), ", ")) -
PARI
{a(n)=(1/n!)*polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*x^k/(1+(3*k+1)^3*x +x*O(x^n))^(k+1)), n)} for(n=0, 20, print1(a(n), ", ")) -
PARI
{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(3*k+1)^(3*n+1))} for(n=0, 20, print1(a(n), ", "))
Formula
a(n) = 1/n! * [x^n] Sum_{k>=0} (3*k+1)^(3*k+1) * x^k / (1 + (3*k+1)^3*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (3*k+1)^(3*n+1).
a(n) ~ n^(2*n+1/2) * 3^(6*n+7/3) / (sqrt(2*Pi*(1-r)) * exp(2*n) * r^(n+1/3) * (3-r)^(2*n+1)), where r = -LambertW(-3*exp(-3)) = 0.1785606278779211... (see A226750 = -r) . - Vaclav Kotesovec, May 13 2014