A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2*k} T(i,k) * C(n-k-1,i).

1, 1, 3, 8, 8, 3, 9, 46, 101, 114, 65, 15, 33, 272, 975, 1935, 2289, 1615, 630, 105, 153, 1796, 9175, 26795, 49474, 60080, 48104, 24535, 7245, 945, 873, 13424, 90255, 353507, 902164, 1582455, 1953272, 1700860, 1025927, 408870, 97020, 10395, 5913

Offset: 1

Author

André F. Labossière, Feb 01 2005

Keywords

• nonn
• tabl

Comments

The coefficients T(i,k) along the i-th columns of the triangle are the consecutive partial sums of those found in table A094216.

Examples

			!7 = 7 + 1*C(7-2,1) + 1*C(7-2,2) + 3*C(7-3,1) + ... + 33*C(7-5,1) + 272*C(7-5,2) + 153*C(7-6,1) = 7 + 5 + 10 + 12 + 8*C(4,2) + 8*C(4,3) + 3*C(4,4) + 9*C(3,1) + 46*C(3,2) + 101*C(3,3) + 66 + 272 + 153 = 7 + 5 + 10 + 12 + 48 + 32 + 3 + 27 + 138 + 101 + 66 + 272 + 153 = 874.
		

Links

• Chris Zheng, Jeffrey Zheng, Triangular Numbers and Their Inherent Properties, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.

Crossrefs

Cf. A094216, A102411, A102412, A101752, A003422, A094638, A008276.