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

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..420 (terms 0..200 from Vincenzo Librandi)

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

Example

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 +...

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 *)

Prog

(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)}
(Maxima)
makelist(sum(abs(stirling1(n, k))*3^k*belln(k), k, 0, n), n, 0, 12); /* Emanuele Munarini, Sep 01 2017 */

Crossrefs

Cf. A000262, A136658, A202824.

Sequence in context: A210670 A193468 A132863 * A212070 A192461 A199682

Adjacent sequences: A202823 A202824 A202825 * A202827 A202828 A202829

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2011

Extensions

Example corrected by Vaclav Kotesovec, Feb 12 2013

Status

approved