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

0,3

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

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

Column k=2 of A293991.

Cf. A004212.

Sequence in context: A153344 A193110 A001930 * A333889 A277395 A012584

Adjacent sequences: A049422 A049423 A049424 * A049426 A049427 A049428

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved