A202826 E.g.f.: exp( 1/(1-x)^3 - 1 ).
1, 3, 21, 195, 2241, 30483, 476469, 8383203, 163532385, 3496040163, 81159271029, 2030708891907, 54427341596769, 1554460972941555, 47097454520401749, 1507969940021725347, 50850987639474121281, 1800630391669594010307, 66775808799868618561365
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 195*x^3/3! + 2241*x^4/4! +... where log(A(x)) = 3*x + 6*x^2 + 10*x^3 + 15*x^4 + 21*x^5 + 28*x^6 +...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..420 (terms 0..200 from Vincenzo Librandi)
Programs
-
Mathematica
CoefficientList[Series[E^(1/(1-x)^3-1), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 12 2013 *) Table[Sum[Abs[StirlingS1[n, k]] 3^k BellB[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Sep 01 2017 *) -
Maxima
makelist(sum(abs(stirling1(n,k))*3^k*belln(k),k,0,n),n,0,12); /* Emanuele Munarini, Sep 01 2017 */
-
PARI
{a(n)=n!*polcoeff(exp(1/(1-x +x*O(x^n))^3-1),n)} -
PARI
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)} {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)} {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) *(-1)^(n-k)*3^k)}
Formula
a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 3^k.
a(n) ~ n! * 1/2*3^(1/8)*exp(sqrt(3*n)/2 -3/4 + (3*n)^(1/4)*(4/3*sqrt(n) + 5/24*sqrt(3)) )/(sqrt(2*Pi)*n^(5/8)) * (1 + 871/2560*(3/n)^(1/4)). - Vaclav Kotesovec, Feb 12 2013
a(n+4) - (4*n+15)*a(n+3) + 6*(n+2)*(n+3)*a(n+2) - 4*(n+1)*(n+2)+(n+3)*a(n+1) + n*(n+1)*(n+2)*(n+3)*a(n) = 0. - Emanuele Munarini, Sep 01 2017
Extensions
Example corrected by Vaclav Kotesovec, Feb 12 2013