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A117434 Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.

%I A117434 #14 Sep 08 2022 08:45:24
%S A117434 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,
%T A117434 0,0,0,56,420,792,429,0,0,0,0,14,420,1980,3003,1430,0,0,0,0,0,210,
%U A117434 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
%N A117434 Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.
%H A117434 G. C. Greubel, <a href="/A117434/b117434.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A117434 Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F A117434 T(n, k) = binomial(k, n-k)*Catalan(k).
%F A117434 Sum_{k=0..n} T(n, k) = A052709(n+1).
%F A117434 Sum_{k=0..floor(n/2)} T(n-k, k) = A115178(n) (upward diagonal sums).
%F A117434 T(n, k) = (-1)^(n+k)*A115179(n, k).
%e A117434 Triangle begins as:
%e A117434   1;
%e A117434   0, 1;
%e A117434   0, 1, 2;
%e A117434   0, 0, 4,  5;
%e A117434   0, 0, 2, 15, 14;
%e A117434   0, 0, 0, 15, 56,  42;
%e A117434   0, 0, 0,  5, 84, 210, 132;
%e A117434   0, 0, 0,  0, 56, 420, 792, 429;
%t A117434 Table[CatalanNumber[k]*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 31 2021 *)
%o A117434 (Magma) [Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 31 2021
%o A117434 (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
%Y A117434 Cf. A000108, A052709, A115178, A115178.
%K A117434 easy,nonn,tabl
%O A117434 0,6
%A A117434 _Paul Barry_, Mar 14 2006