A202995 - OEIS

%I #18 Nov 24 2024 20:40:23

%S 1,1,1,1,3,1,0,10,6,1,-4,30,40,10,1,0,36,270,110,15,1,120,-420,1596,

%T 1260,245,21,1,0,-2400,5040,14056,4200,476,28,1,-12096,30240,-46080,

%U 136080,72576,11340,840,36,1,0,423360,-756000,795600,1197000,276192,26460,1380,45,1

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

%C Triangle T(n,m)*m!/((n-m)!*n!)=

%C 1. Riordan Array (1,x^2*exp(x)/(exp(x)-1)) without first column.

%C 2. Riordan Array (x*exp(x)/(exp(x)-1),x^2*exp(x)/(exp(x)-1)) numbering triangle (0,0).

%F T(n,m):=(n!*(n-m)!*sum(j=0..m, (j!*binomial(m,j)*sum(k=0..n-2*m+j, (k!*stirling1(k+j,j)*stirling2(n-2*m+j,k))/((n-2*m+j)!*(k+j)!)))))/m!.

%e 1,

%e 1, 1,

%e 1, 3, 1,

%e 0, 10, 6, 1,

%e -4, 30, 40, 10, 1,

%e 0, 36, 270, 110, 15, 1,

%e 120, -420, 1596, 1260, 245, 21, 1

%o (Maxima) T(n,m):=(n!*(n-m)!*sum((j!*binomial(m,j)*sum((k!*stirling1(k+j,j)*stirling2(n-2*m+j,k))/((n-2*m+j)!*(k+j)!),k,0,n-2*m+j)),j,0,m))/m!;

%K sign,tabl,changed

%O 1,5

%A _Vladimir Kruchinin_, Dec 27 2011