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).

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, 20, 56, 162, 508, 1476, 0, 1, 7, 27, 85, 269, 939, 3549, 12123, 0, 1, 8, 35, 122, 415, 1540, 6413, 28498, 111866, 0, 1, 9, 44, 168, 609, 2361, 10314, 50382, 257922, 1143554, 0, 1

Offset: 0

Author

Paul D. Hanna, Dec 17 2003

Keywords

• nonn
• tabl

Comments

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).

Examples

			Rows begin:
{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},
{1,6,20,56,162,508,1476,0},
{1,7,27,85,269,939,3549,12123,0},...
Initial terms of the binomial transform of each row forms A059438:
{1},
{1,1},
{1,2,3},
{1,3,7,13},
{1,4,12,32,71},
{1,5,18,58,177,461},
{1,6,25,92,327,1142,3447},
{1,7,33,135,531,2109,8411,29093},
{1,8,42,188,800,3440,15366,69692,273343},...
which has row sums equal to the factorials.
		

Crossrefs

Cf. A075834, A059438, A003319.