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%I #23 Mar 28 2020 14:04:17
%S 1,0,1,6,0,1,-24,24,0,1,170,-120,60,0,1,-1320,1380,-360,120,0,1,11816,
%T -14280,6090,-840,210,0,1,-118944,171808,-77280,19600,-1680,336,0,1,
%U 1329156,-2249856,1181376,-292320,51660,-3024,504,0,1,-16313760,32093280
%N Triangle T(n,m) = coefficient of x^n in expansion of x^m*(x+1)^(log(1+x)*m) = 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,2*k)*(2*k)!/k! where S1 are the Stirling cycle numbers A132393. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 15 2016
%H Indranil Ghosh, <a href="/A202185/b202185.txt">Table of n, a(n) for n = Rows 1..50, flattened</a>
%F T(n,m) = binomial(n,m)*sum(k=0..n-m, ((2*k)!*m^k*stirling1(n-m,2*k))/k!).
%e 1,
%e 0, 1,
%e 6, 0, 1,
%e -24, 24, 0, 1,
%e 170, -120, 60, 0, 1,
%e -1320, 1380, -360, 120, 0, 1,
%e 11816, -14280, 6090, -840, 210, 0, 1
%t Flatten[Table[Binomial[n,m]*Sum[((2k)!*m^k*StirlingS1[n-m,2k])/k!,{k,0,n-m}],{n,1,7},{m,1,n}]] (* _Indranil Ghosh_, Feb 21 2017 *)
%o (Maxima)
%o T(n,m):=binomial(n,m)*sum(((2*k)!*m^k*stirling1(n-m,2*k))/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 A202185_row(n):
%o f = lambda n: (-1)^n*(n+1)*sum(factorial(2*k)*stirling_number1(n,2*k)/ factorial(k) for k in (0..n))
%o return bell_transform(n, [f(k) for k in (0..n)])
%o [A202185_row(n) for n in (0..9)] # _Peter Luschny_, Jan 15 2016
%o (PARI) T(n, m) = binomial(n,m)*sum(k=0,n-m, ((2*k)!*m^k*stirling(n-m,2*k,1))/k!); \\ _Michel Marcus_, Jan 16 2016
%K sign,tabl
%O 1,4
%A _Vladimir Kruchinin_, Dec 13 2011
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