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A112500
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Triangle of column sequences with a certain o.g.f. pattern.
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6
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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; internal format)
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OFFSET
| 0,5
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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).
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LINKS
| W. Lang, First ten rows.
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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.
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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).
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CROSSREFS
| Cf. A112501 (row sums).
Sequence in context: A147564 A090981 A087903 * A152938 A154096 A146898
Adjacent sequences: A112497 A112498 A112499 * A112501 A112502 A112503
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 14 2005
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