OFFSET
1,5
COMMENTS
1. Expansion of (x*Bernoulli(x)^m=x^m+sum(n>m m!*sum(k=1..n-m, (k!*stirling1(m+k,m)*stirling2(n-m,k))/(m+k)!))/(n-m)!*x^n)
2. Riordan Array (1,x*Bernoulli(x)) without first column.
3. Riordan Array (Bernoulli(x),x*Bernoulli(x)) numbering triangle (0,0).
LINKS
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
Dmitry V. Kruchinin and Vladimir V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
FORMULA
T(n,m)=n!*sum(k=0..n-m, (k!*stirling1(m+k,m)*stirling2(n-m,k))/(m+k)!).
T(n,m):=n!*(n-m)!/m!*sum(k=0..n-m, k!*binomial(m+k-1,m-1)*sum(j=0..k, ((-1)^j*stirling2(n-m+j,j))/((k-j)!*(n-m+j)!))). [Vladimir Kruchinin, Jun 14 2013 ]
EXAMPLE
1,
-1,1,
1,-3,1,
0,10,-6,1,
-4,-30,40,-10,1,
0,36,-270,110,-15,1,
120,420,1596,-1260,245,-21,1
MAPLE
A191578 := proc(n, m)
if m=n then
1;
else
add(combinat[stirling2] (n-m, k) *k! *combinat[stirling1](m+k, m)/(m+k)!, k=1..n-m) ;
%*n! ;
end if;
end proc: # R. J. Mathar, Jun 14 2013
MATHEMATICA
t[n_, m_] := n!*Sum[ (k!*StirlingS1[m+k, m]*StirlingS2[n-m, k])/(m+k)!, {k, 1, n-m}]; t[n_, n_] = 1; Table[t[n, m], {n, 1, 12}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 22 2013 *)
PROG
(Maxima)
T(n, m):=n!*sum((k!*stirling1(m+k, m)*stirling2(n-m, k))/(m+k)!, k, 0, n-m); /* Vladimir Kruchinin, Jun 14 2013 */
(Maxima)
T(n, m):=n!*(n-m)!/m!*sum(k!*binomial(m+k-1, m-1)*sum(((-1)^j*stirling2(n-m+j, j))/((k-j)!*(n-m+j)!), j, 0, k), k, 0, n-m); /* Vladimir Kruchinin, Jun 14 2013 */
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Vladimir Kruchinin, Jun 07 2011
STATUS
approved