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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).

%I A191578 #29 Nov 09 2024 14:48:50
%S A191578 1,-1,1,1,-3,1,0,10,-6,1,-4,-30,40,-10,1,0,36,-270,110,-15,1,120,420,
%T A191578 1596,-1260,245,-21,1,0,-2400,-5040,14056,-4200,476,-28,1,-12096,
%U A191578 -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
%N 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).
%C A191578 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)
%C A191578 2. Riordan Array (1,x*Bernoulli(x)) without first column.
%C A191578 3. Riordan Array (Bernoulli(x),x*Bernoulli(x)) numbering triangle (0,0).
%H A191578 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%H A191578 Dmitry V. Kruchinin and Vladimir V. Kruchinin, <a href="http://www.emis.de/journals/JIS/VOL18/Kruchinin/kruch9.pdf">A Generating Function for the Diagonal T_{2n,n} in Triangles</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
%F A191578 T(n,m)=n!*sum(k=0..n-m, (k!*stirling1(m+k,m)*stirling2(n-m,k))/(m+k)!).
%F A191578 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 ]
%e A191578 1,
%e A191578 -1,1,
%e A191578 1,-3,1,
%e A191578 0,10,-6,1,
%e A191578 -4,-30,40,-10,1,
%e A191578 0,36,-270,110,-15,1,
%e A191578 120,420,1596,-1260,245,-21,1
%p A191578 A191578 := proc(n, m)
%p A191578     if m=n then
%p A191578         1;
%p A191578     else
%p A191578         add(combinat[stirling2] (n-m, k) *k! *combinat[stirling1](m+k, m)/(m+k)!, k=1..n-m) ;
%p A191578         %*n! ;
%p A191578     end if;
%p A191578 end proc: # _R. J. Mathar_, Jun 14 2013
%t A191578 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 *)
%o A191578 (Maxima)
%o A191578 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 */
%o A191578 (Maxima)
%o A191578 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 */
%Y A191578 First column T(n,1)=A129814(n-1)
%K A191578 sign,tabl
%O A191578 1,5
%A A191578 _Vladimir Kruchinin_, Jun 07 2011