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%I #22 Mar 28 2020 14:04:05
%S 1,0,1,6,0,1,-12,24,0,1,100,-60,60,0,1,-540,960,-180,120,0,1,4158,
%T -6300,4620,-420,210,0,1,-33600,71904,-35280,15680,-840,336,0,1,
%U 310896,-725760,557928,-136080,42840,-1512,504,0,1,-3160080,8723520,-6652800
%N Triangle T(n,m) = coefficient of x^n in expansion of x^m*(x+1)^(m*x) = sum(n>=m, T(n,m) x^n*m!/n!).
%C Also the Bell transform of (-1)^n*(n+1)!*Sum_{k=0..n} S1(n-k,k)/(n-k)! where S1 are the Stirling cycle numbers A132393. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 15 2016
%H Robert Israel, <a href="/A202183/b202183.txt">Table of n, a(n) for n = 1..10011</a> (rows 1 to 141, flattened)
%F T(n,m) = n!/m!*sum(k=0..n-m, (m^k*stirling1(n-m-k,k))/(n-m-k)!).
%e 1,
%e 0, 1,
%e 6, 0, 1,
%e -12, 24, 0, 1,
%e 100, -60, 60, 0, 1,
%e -540, 960, -180, 120, 0, 1,
%e 4158, -6300, 4620, -420, 210, 0, 1
%p N:= 10: # for rows 1 to N
%p for m from 1 to N do
%p S[m]:= series(x^m*(x+1)^(m*x),x,N+1);
%p od:
%p seq(seq(coeff(S[m],x,n)*n!/m!,m=1..n),n=1..N);# _Robert Israel_, Jan 15 2016
%o (Maxima)
%o T(n,m):=n!/m!*sum((m^k*stirling1(n-m-k,k))/(n-m-k)!,k,0,n-m);
%o (Sage) # uses[bell_transform from A264428]
%o # Adds a column 1,0,0,0,... at the left side of the triangle.
%o def A202183_row(n):
%o f = lambda n: (-1)^n*factorial(n+1)*sum(stirling_number1(n-k, k)/factorial(n-k) for k in (0..n))
%o return bell_transform(n, [f(k) for k in (0..n)])
%o [A202183_row(n) for n in (0..9)] # _Peter Luschny_, Jan 15 2016
%o (PARI) T(n,m) = n!/m!*sum(k=0, n-m, (m^k*stirling(n-m-k,k,1))/(n-m-k)!); \\ _Michel Marcus_, Jan 16 2016
%K sign,tabl
%O 1,4
%A _Vladimir Kruchinin_, Dec 13 2011
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