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A112500 Triangle of column sequences with a certain o.g.f. pattern. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Oct 14 2005
STATUS
approved

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Last modified April 19 02:28 EDT 2024. Contains 371782 sequences. (Running on oeis4.)