OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..616 (terms 0..200 from Vincenzo Librandi)
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, J. Integer Seqs., Vol. 20 (2017), #17.8.2.
FORMULA
E.g.f.: exp(x+x^2+(x^3)/3).
a(n) = n! * sum(k=0..n, sum(j=0..k, binomial(3*j,n) * (-1)^(k-j)/(3^k * (k-j)!*j!))). [Vladimir Kruchinin, Feb 07 2011]
Conjecture: -a(n) +a(n-1) +(2*n-2)*a(n-2) + (2-3*n+n^2)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ exp(n^(2/3)+n^(1/3)/3-2*n/3-2/9)*n^(2*n/3)/sqrt(3)*(1+59/(162*n^(1/3))). - Vaclav Kotesovec, Oct 08 2012
From Emanuele Munarini, Oct 20 2014: (Start)
Recurrence: a(n+3) = a(n+2)+2*(n+2)*a(n+1)+(n+2)*(n+1)*a(n).
It derives from the differential equation for the e.g.f.: A'(x) = (1+2*x+x^2)*A(x).
So, the above conjecture is true.
b(n) = a(n+1) = sum((n!/k!)*sum(bin(k,i)*bin(k-i+2,n-2*i-k)/3^i,i=0..k),k=0..n).
E.g.f. for b(n) = a(n+1): (1+t)^2*exp(t+t^2+t^3/3).
(End)
a(n) = Sum_{k=0..n} Stirling1(n,k) * A004212(k). - Seiichi Manyama, Jan 31 2024
a(n) = (1/exp(1/3)) * n! * Sum_{k>=0} binomial(3*k,n)/(3^k * k!). - Seiichi Manyama, Jan 18 2025
MATHEMATICA
Table[n!*SeriesCoefficient[E^(x+x^2+(x^3)/3), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)
PROG
(PARI) x='x+O('x^66); Vec(serlaplace(exp(x+x^2+(x^3)/3))) \\ Joerg Arndt, May 04 2013
(Maxima) /* for b(n) = a(n+1) */
b(n) := sum((n!/k!)*sum(binomial(k, i)*binomial(k-i+2, n-2*i-k)/3^i, i, 0, k), k, 0, n);
makelist(b(n), n, 0, 24); /* Emanuele Munarini, Oct 20 2014 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved