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%I #10 Sep 08 2022 08:45:23
%S 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,
%T 132,0,0,0,0,-56,420,-792,429,0,0,0,0,14,-420,1980,-3003,1430,0,0,0,0,
%U 0,210,-2640,9009,-11440,4862,0,0,0,0,0,-42,1980,-15015,40040,-43758,16796
%N Expansion of c(x*y*(1-x)), c(x) the g.f. of A000108.
%C Since C(x*(1-x)) = 1/(1-x), the row sums of this triangle are (1,1,1,...). This establishes the identity Sum_{k=0..n} (-1)^(n-k)*C(k)*binomial(k,n-k)} = 1, where C(x) is the g.f. of A000178.
%H G. C. Greubel, <a href="/A115179/b115179.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = (-1)^(n-k)*binomial(k, n-k)*Catalan(k).
%F Sum_{k=0..n} T(n, k) = A000012(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A115178(n) (upward diagonal sums).
%F T(n, k) = (-1)^(n+k)*A117434(n, k).
%e Triangle begins
%e 1;
%e 0, 1;
%e 0, -1, 2;
%e 0, 0, -4, 5;
%e 0, 0, 2, -15, 14;
%e 0, 0, 0, 15, -56, 42;
%e 0, 0, 0, -5, 84, -210, 132;
%e 0, 0, 0, 0, -56, 420, -792, 429;
%t Table[(-1)^(n+k)*CatalanNumber[k]*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 31 2021 *)
%o (Magma) [(-1)^(n+k)*Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 31 2021
%o (Sage) flatten([[(-1)^(n+k)*binomial(k, n-k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 31 2021
%Y Cf. A000012, A000178, A115178, A117434.
%K easy,sign,tabl
%O 0,6
%A _Paul Barry_, Mar 14 2006
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