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A027478 Triangle of the cube of the normalized, unsigned Stirling matrix of the first kind.

%I A027478 #14 May 07 2025 14:34:49
%S A027478 1,7,1,176,39,1,10746,2951,126,1,1297704,407450,22535,310,1,272866980,
%T A027478 94128364,6139575,112435,645,1,91570835040,33910601508,2487385684,
%U A027478 54814095,426475,1197,1,46034917019280,18030534782364,1446119232796,36402686929,345710680,1333906,2044,1
%N A027478 Triangle of the cube of the normalized, unsigned Stirling matrix of the first kind.
%C A027478 The absolute values are unchanged if one uses the signed Stirling numbers of the first kind.
%F A027478 Let A be the lower triangular matrix with entries a[ i, j ] = (-1)^(i+j)*s(i, j)/i! if j<=i, 0 if j>i, where s(i,j) is the Stirling number of the first kind. Let N be the column vector ((i!^3)).
%F A027478 T is the lower triangular matrix  A.A.A.N.
%e A027478 The first rows of the triangle are :
%e A027478   1,
%e A027478   7, 1,
%e A027478   176, 39, 1,
%e A027478   10746, 2951, 126, 1,
%e A027478   1297704, 407450, 22535, 310, 1,
%e A027478   272866980, 94128364, 6139575, 112435, 645, 1,
%e A027478   ...
%t A027478 Module[{nmax=8,m},m=(Table[Table[(-1)^(i+j) StirlingS1[i,j]/i!,{j,1,nmax}],{i,1,nmax}]);m=m.m.m*Table[i!^3,{i,1,nmax}]; Flatten[Table[Table[m[[i,j]],{j,1,i}],{i,1,nmax}],1]]
%Y A027478 Cf. A027477 for the quadratic version.
%Y A027478 Cf. A027479 for the quartic version.
%Y A027478 Cf. A027482 is the first subdiagonal of this triangle.
%K A027478 nonn,tabl,easy
%O A027478 1,2
%A A027478 _Olivier Gérard_
%E A027478 Definition, formula and program edited for clarity by _Olivier Gérard_, Jan 20 2019