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A180957 Generalized Narayana triangle for (-1)^n.

%I A180957 #7 Apr 06 2021 23:09:39
%S A180957 1,1,1,1,1,1,1,0,0,1,1,-2,-5,-2,1,1,-5,-15,-15,-5,1,1,-9,-30,-41,-30,
%T A180957 -9,1,1,-14,-49,-77,-77,-49,-14,1,1,-20,-70,-112,-125,-112,-70,-20,1,
%U A180957 1,-27,-90,-126,-117,-117,-126,-90,-27,1,1,-35,-105,-90,45,131,45,-90,-105,-35,1
%N A180957 Generalized Narayana triangle for (-1)^n.
%H A180957 G. C. Greubel, <a href="/A180957/b180957.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A180957 G.f.: 1/(1 -x -x*y + x/(1 -x -x*y)) = (1 -x*(1+y))/(1 -2*x*(1+y) +x^2*(1 +3*y +y^2)).
%F A180957 E.g.f.: exp((1+y)*x) * cos(sqrt(y)*x).
%F A180957 T(n, k) = Sum_{j=0..n} (-1)^(k-j)*binomial(n,j)*binomial(n-j, 2*(k-j)).
%F A180957 Sum_{k=0..n} T(n, k) = A139011(n) (row sums).
%F A180957 Sum_{k=0..floor(n/2)} T(n-k, k) = A180958(n) (diagonal sums).
%e A180957 Triangle begins
%e A180957   1;
%e A180957   1,   1;
%e A180957   1,   1,    1;
%e A180957   1,   0,    0,    1;
%e A180957   1,  -2,   -5,   -2,    1;
%e A180957   1,  -5,  -15,  -15,   -5,    1;
%e A180957   1,  -9,  -30,  -41,  -30,   -9,    1;
%e A180957   1, -14,  -49,  -77,  -77,  -49,  -14,   1;
%e A180957   1, -20,  -70, -112, -125, -112,  -70, -20,    1;
%e A180957   1, -27,  -90, -126, -117, -117, -126, -90,  -27,   1;
%e A180957   1, -35, -105,  -90,   45,  131,   45, -90, -105, -35, 1;
%t A180957 T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j,0,n}];
%t A180957 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 06 2021 *)
%o A180957 (Magma)
%o A180957 A180957:= func< n,k | (&+[ (-1)^(k-j)*Binomial(n, j)*Binomial(n-j, 2*(k-j)) : j in [0..n]]) >;
%o A180957 [A180957(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Apr 06 2021
%o A180957 (Sage)
%o A180957 def A180957(n,k): return sum( (-1)^(k+j)*binomial(n,j)*binomial(n-j, 2*(k-j)) for j in (0..n))
%o A180957 flatten([[A180957(n,k) for k in (0..n)] for n in [0..15]]) # _G. C. Greubel_, Apr 06 2021
%Y A180957 Cf. A056241, A139011.
%Y A180957 Variant: A061176.
%K A180957 easy,sign,tabl
%O A180957 0,12
%A A180957 _Paul Barry_, Sep 28 2010