A115179 - OEIS

%I #14 Sep 11 2024 00:29:44

%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} T(n, k) = Sum_{k=0..n} (-1)^(n-k)*A000108(k)*binomial(k,n-k) = 1.

%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, A000108, A115178, A117434.

%K easy,sign,tabl

%O 0,6

%A _Paul Barry_, Mar 14 2006