A191578 Triangle read by rows, based on expansion of (x^2/(exp(x)-1))^m = x^m+sum(n>m T(n,m)*m!/((n-m)!*n!)*x^n).

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, 0, -2400, -5040, 14056, -4200, 476, -28, 1, -12096, -30240, -46080, -136080, 72576, -11340, 840, -36, 1, 0, 423360, 756000, 795600, -1197000, 276192, -26460, 1380, -45, 1, 3024000, 5987520, 4213440, 6098400, 17087400, -6652800, 857472, -55440, 2145, -55, 1, 0, -163296000, -251475840, -220651200, -158004000, 151169040, -27941760, 2297592, -106920, 3190, -66, 1

Offset: 1

Author

Vladimir Kruchinin, Jun 07 2011

Keywords

• sign
• tabl

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).

Examples

			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
		

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.

Crossrefs

First column T(n,1)=A129814(n-1)

Programs

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 *)

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 */

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 ]