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

0,5

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).

Links

Table of n, a(n) for n=0..55.

W. Lang, First ten rows.

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<k.

Example

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).

Crossrefs

Cf. A112501 (row sums).

Sequence in context: A090981 A087903 A287532 * A152938 A154096 A146898

Adjacent sequences: A112497 A112498 A112499 * A112501 A112502 A112503

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 14 2005

Status

approved