A091063 - OEIS

%I #3 Mar 30 2012 18:36:39

%S 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 20,56,162,508,1476,0,1,7,27,85,269,939,3549,12123,0,1,8,35,122,415,

%U 1540,6413,28498,111866,0,1,9,44,168,609,2361,10314,50382,257922,1143554,0,1

%N 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 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 Rows begin:

%e {1},

%e {1,0},

%e {1,1,0},

%e {1,2,2,0},

%e {1,3,5,7,0},

%e {1,4,9,18,34,0},

%e {1,5,14,34,86,206,0},

%e {1,6,20,56,162,508,1476,0},

%e {1,7,27,85,269,939,3549,12123,0},...

%e Initial terms of the binomial transform of each row forms A059438:

%e {1},

%e {1,1},

%e {1,2,3},

%e {1,3,7,13},

%e {1,4,12,32,71},

%e {1,5,18,58,177,461},

%e {1,6,25,92,327,1142,3447},

%e {1,7,33,135,531,2109,8411,29093},

%e {1,8,42,188,800,3440,15366,69692,273343},...

%e which has row sums equal to the factorials.

%Y Cf. A075834, A059438, A003319.

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Dec 17 2003