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A091063 Triangle, read by rows, such that the initial terms of the binomial transform of the n-th row forms the n-th row of triangle A059438 transposed (permutations of [1..n] with k components).

%I A091063 #3 Mar 30 2012 18:36:39
%S A091063 1,1,0,1,1,0,1,2,2,0,1,3,5,7,0,1,4,9,18,34,0,1,5,14,34,86,206,0,1,6,
%T A091063 20,56,162,508,1476,0,1,7,27,85,269,939,3549,12123,0,1,8,35,122,415,
%U A091063 1540,6413,28498,111866,0,1,9,44,168,609,2361,10314,50382,257922,1143554,0,1
%N A091063 Triangle, read by rows, such that the initial terms of the binomial transform of the n-th row forms the n-th row of triangle A059438 transposed (permutations of [1..n] with k components).
%C A091063 The main diagonal equals A075834 shift 1 place left; subsequent diagonals of this triangle are self-convolutions of the main diagonal. A075834 has the property that the n-th term of the n-th self-convolution of A075834 equals n!. The first (n+1) terms of the binomial transform of the n-th row forms the n-th row of triangle A059438 transposed, which has row sums equal to the factorials. A059438 is also formed from the self-convolutions of its main diagonal (A003319).
%e A091063 Rows begin:
%e A091063 {1},
%e A091063 {1,0},
%e A091063 {1,1,0},
%e A091063 {1,2,2,0},
%e A091063 {1,3,5,7,0},
%e A091063 {1,4,9,18,34,0},
%e A091063 {1,5,14,34,86,206,0},
%e A091063 {1,6,20,56,162,508,1476,0},
%e A091063 {1,7,27,85,269,939,3549,12123,0},...
%e A091063 Initial terms of the binomial transform of each row forms A059438:
%e A091063 {1},
%e A091063 {1,1},
%e A091063 {1,2,3},
%e A091063 {1,3,7,13},
%e A091063 {1,4,12,32,71},
%e A091063 {1,5,18,58,177,461},
%e A091063 {1,6,25,92,327,1142,3447},
%e A091063 {1,7,33,135,531,2109,8411,29093},
%e A091063 {1,8,42,188,800,3440,15366,69692,273343},...
%e A091063 which has row sums equal to the factorials.
%Y A091063 Cf. A075834, A059438, A003319.
%K A091063 nonn,tabl
%O A091063 0,8
%A A091063 _Paul D. Hanna_, Dec 17 2003