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A049425 Row sums of triangle A049404. 8
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 (list; graph; refs; listen; history; text; internal format)
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)

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.

Sequence in context: A153344 A193110 A001930 * A277395 A012584 A101899

Adjacent sequences:  A049422 A049423 A049424 * A049426 A049427 A049428

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified May 25 05:30 EDT 2018. Contains 304541 sequences. (Running on oeis4.)