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A117434 Expansion of c(x*y(1+x)), c(x) the g.f. of A000108. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
FORMULA
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).
EXAMPLE
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;
MATHEMATICA
Table[CatalanNumber[k]*Binomial[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *)
PROG
(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
CROSSREFS
Sequence in context: A094295 A085969 A115179 * A131742 A257813 A278280
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Mar 14 2006
STATUS
approved

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Last modified April 22 14:59 EDT 2024. Contains 371905 sequences. (Running on oeis4.)