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A117434
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Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.
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5
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1, 0, 1, 0, 1, 2, 0, 0, 4, 5, 0, 0, 2, 15, 14, 0, 0, 0, 15, 56, 42, 0, 0, 0, 5, 84, 210, 132, 0, 0, 0, 0, 56, 420, 792, 429, 0, 0, 0, 0, 14, 420, 1980, 3003, 1430, 0, 0, 0, 0, 0, 210, 2640, 9009, 11440, 4862, 0, 0, 0, 0, 0, 42, 1980, 15015, 40040, 43758, 16796, 0, 0, 0, 0, 0, 0, 792, 15015, 80080, 175032, 167960, 58786
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OFFSET
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0,6
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LINKS
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FORMULA
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T(n, k) = binomial(k, n-k)*Catalan(k).
Sum_{k=0..n} T(n, k) = A052709(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A115178(n) (upward diagonal sums).
T(n, k) = (-1)^(n+k)*A115179(n, k).
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EXAMPLE
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Triangle begins as:
1;
0, 1;
0, 1, 2;
0, 0, 4, 5;
0, 0, 2, 15, 14;
0, 0, 0, 15, 56, 42;
0, 0, 0, 5, 84, 210, 132;
0, 0, 0, 0, 56, 420, 792, 429;
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MATHEMATICA
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Table[CatalanNumber[k]*Binomial[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *)
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PROG
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(Magma) [Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 31 2021
(Sage) flatten([[binomial(k, n-k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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