A049425 Row sums of triangle A049404.
1, 1, 3, 9, 33, 141, 651, 3333, 18369, 108153, 678771, 4495041, 31324833, 228803589, 1744475643, 13852095741, 114235118721, 976176336753, 8627940414819, 78726234866553, 740440277799201, 7168107030092541, 71331617341611243, 728811735008913909, 7637128289949856833, 81995144342947130601
Offset: 0
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.
Programs
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Mathematica
Table[n!*SeriesCoefficient[E^(x+x^2+(x^3)/3),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *) -
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 */
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PARI
x='x+O('x^66); Vec(serlaplace(exp(x+x^2+(x^3)/3))) \\ Joerg Arndt, May 04 2013
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