A112500 Triangle of column sequences with a certain o.g.f. pattern.

1, 1, 1, 1, 4, 1, 1, 11, 10, 1, 1, 26, 60, 20, 1, 1, 57, 282, 225, 35, 1, 1, 120, 1149, 1882, 665, 56, 1, 1, 247, 4272, 13070, 9107, 1666, 84, 1, 1, 502, 14932, 79872, 100751, 35028, 3696, 120, 1, 1, 1013, 49996, 444902, 957197, 584325, 113428, 7470, 165, 1, 1

Offset: 0

Author

Wolfdieter Lang, Oct 14 2005

Keywords

• nonn
• easy
• tabl

Comments

The column o.g.f.s of this triangle appear as factors in the column o.g.f.s of triangle A008517 (second-order Eulerian numbers).

Examples

			Rows: [1]; [1,1]; [1,4,1]; [1,11,10,1]; [1,26,60,20,1]; [1,57,282,225,35,1]; ...
a(4,3)= 60 = 6 + 12 + 9 + 12 + 9 + 12 from the binomial(4,2)=6 terms of the sum corresponding to (n_1,n_2,n_3) = (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1) and (0,1,1).
		

Links

• W. Lang, First ten rows.

Crossrefs

Cf. A112501 (row sums).

Formula

G.f. column k: G(k, x):= x^(k-1)/product((1-j*x)^(k-j+1), j=1..k), k>=1.

The column sequences start with A000012 (powers of 1), A000295 (Eulerian numbers), A112502-A112504.

a(n+k-1, k)=sum of product(binomial(n_j + k - 1, k - 1)*j^(n_j), j=1..k) with sum(n_j, j=1..k)=n, n_j >=0. There are binomial(n+k-1, k-1) terms of this sum and 1<=k<=n+1. a(n, k)=0 if n+1

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