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%I #23 Mar 27 2020 12:19:30
%S 1,0,1,6,0,1,12,24,0,1,-20,60,60,0,1,540,240,180,120,0,1,-882,6300,
%T 2100,420,210,0,1,-6720,-8736,35280,8960,840,336,0,1,189936,181440,
%U 13608,136080,27720,1512,504,0,1
%N Triangle T(n,k) = coefficient of x^n in expansion of x^k*(1+x+x^2)^(k*x) = sum(n>=k, T(n,k) x^n*k!/n!).
%C Also the Bell transform of (n+1)!*Sum_{k=0..n}(Sum_{i=k..n-k}((-1)^(i-k)*S1(i,k)* binomial(i,n-k-i)/i!) where S1 are the Stirling cycle numbers A132393. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 15 2016
%F T(n,m) = n!/m!*sum(k=0..n-m, (sum(i=k,n-m-k, (stirling1(i,k)*binomial(i,n-m-k-i))/i!))*m^k).
%e 1
%e 0, 1,
%e 6, 0, 1,
%e 12, 24, 0, 1,
%e -20, 60, 60, 0, 1,
%e 540, 240, 180, 120, 0, 1,
%e -882, 6300, 2100, 420, 210, 0, 1]
%t Table[n!/m! Sum[Sum[(StirlingS1[i, k] Binomial[i, n - m - k - i])/i!, {i, k, n - m - k}] m^k, {k, 0, n - m}], {n, 9}, {m, n}] // Flatten (* _Michael De Vlieger_, Jan 15 2016 *)
%o (Maxima)
%o T(n,m):=n!/m!*sum((sum((stirling1(i,k)*binomial(i,n-m-k-i))/i!,i,k,n-m-k))*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 A202189_row(n):
%o f = lambda n: factorial(n+1)*sum(sum((-1)^(i-k)*stirling_number1(i,k)* binomial(i,n-k-i)/factorial(i) for i in (k..n-k)) for k in (0..n))
%o return bell_transform(n, [f(k) for k in (0..n)])
%o [A202189_row(n) for n in (0..9)] # _Peter Luschny_, Jan 15 2016
%K sign,tabl
%O 1,4
%A _Vladimir Kruchinin_, Dec 13 2011
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