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A104259
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Triangle T read by rows: matrix product of Pascal and Catalan triangle.
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8
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1, 2, 1, 5, 4, 1, 15, 14, 6, 1, 51, 50, 27, 8, 1, 188, 187, 113, 44, 10, 1, 731, 730, 468, 212, 65, 12, 1, 2950, 2949, 1956, 970, 355, 90, 14, 1, 12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1, 51822, 51821, 35643, 19474, 8612, 3021, 805, 152, 18, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also, Riordan array (G,G), G(t)=(1 - ((1-5*t)/(1-t))^(1/2))/(2*t).
From Emanuele Munarini, May 18 2011: (Start)
Row sums = A002212.
Diagonal sums = A190737.
Central coefficients = A190738. (End)
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LINKS
| D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees
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FORMULA
| T(n,k) = sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i=k..n).
T(n+1,k+2) = T(n+1,k+1) + T(n,k+2) - T(n,k+1) - T(n,k). [Emanuele Munarini, May 18 2011]
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EXAMPLE
| Triangle begins:
1
2, 1
5, 4, 1
15, 14, 6, 1
51, 50, 27, 8, 1
188, 187, 113, 44, 10, 1
731, 730, 468, 212, 65, 12, 1
2950, 2949, 1956, 970, 355, 90, 14, 1
12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1
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MATHEMATICA
| Flatten[Table[Sum[Binomial[n, i]Binomial[2i-k, i-k](k+1)/(i+1), {i, k, n}], {n, 0, 100}, {k, 0, n}]] [Emanuele Munarini, May 18 2011]
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PROG
| (Maxima) create_list(sum(binomial(n, i)*binomial(2*i-k, i-k)*(k+1)/(i+1), i, k, n), n, 0, 12, k, 0, n); [Emanuele Munarini, May 18 2011]
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CROSSREFS
| T = A007318 * A033184.
Left-hand columns include A007317, A007317 - 1. Row sums are in A002212.
Sequence in context: A193673 A126181 A154930 * A137650 A171515 A110271
Adjacent sequences: A104256 A104257 A104258 * A104260 A104261 A104262
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KEYWORD
| nonn,tabl
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AUTHOR
| Ralf Stephan, Mar 17 2005
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