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A112519 Riordan array (1, x*c(x)*c(-x*c(x))), c(x) the g.f. of A000108.

%I A112519 #12 Sep 08 2022 08:45:22
%S A112519 1,0,1,0,0,1,0,2,0,1,0,1,4,0,1,0,12,2,6,0,1,0,14,28,3,8,0,1,0,100,32,
%T A112519 48,4,10,0,1,0,180,249,54,72,5,12,0,1,0,990,440,455,80,100,6,14,0,1,0,
%U A112519 2310,2552,792,726,110,132,7,16,0,1,0,10920,5876,4836,1248,1070,144,168,8,18,0,1
%N A112519 Riordan array (1, x*c(x)*c(-x*c(x))), c(x) the g.f. of A000108.
%C A112519 Row sums are A112520. Second column is essentially A055392. Inverse is A112517. Riordan array product (1, x*c(x))*(1, x*c(-x)).
%H A112519 G. C. Greubel, <a href="/A112519/b112519.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A112519 Riordan array (1, (sqrt(3-2*sqrt(1-4*x)) - 1)/2).
%F A112519 T(n, k) = (k/n)*Sum_{j=0..n} (-1)^(j-k)*C(2*n-j-1, n-j)*C(2*j-k-1, j-k), with T(0, 0) = 1.
%F A112519 T(n, k) = (k/n)*binomial(2*n-k-1, n-1)*Hypergoemetric3F2([k-n, k/2, (1+k)/2], [k-2*n+1, k], -4), with T(0, 0) = 1. - _G. C. Greubel_, Jan 12 2022
%e A112519 Triangle begins
%e A112519   1;
%e A112519   0,   1;
%e A112519   0,   0,   1;
%e A112519   0,   2,   0,   1;
%e A112519   0,   1,   4,   0,  1;
%e A112519   0,  12,   2,   6,  0,   1;
%e A112519   0,  14,  28,   3,  8,   0,  1;
%e A112519   0, 100,  32,  48,  4,  10,  0,  1;
%e A112519   0, 180, 249,  54, 72,   5, 12,  0, 1;
%e A112519   0, 990, 440, 455, 80, 100,  6, 14, 0, 1;
%t A112519 (* First program *)
%t A112519 c[x_]:= (1 - Sqrt[1-4x])/(2x);
%t A112519 (* The function RiordanArray is defined in A256893. *)
%t A112519 RiordanArray[1&, # c[#] c[-# c[#]]&, 12] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%t A112519 (* Second program *)
%t A112519 T[n_, k_]:= If[k==n, 1, (k/n)*Binomial[2*n-k-1, n-1]*HypergeometricPFQ[{k-n, k/2, (1+k)/2}, {k-2*n+1, k}, -4]];
%t A112519 Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Jan 12 2022 *)
%o A112519 (Magma)
%o A112519 A112519:= func< n,k | n eq 0 and k eq 0 select 1 else (k/n)*(&+[(-1)^j*Binomial(2*n-k-j-1, n-k-j)*Binomial(2*j+k-1, j): j in [0..n-k]]) >;
%o A112519 [A112519(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 12 2022
%o A112519 (Sage)
%o A112519 @CachedFunction
%o A112519 def A112519(n,k):
%o A112519     if (k==n): return 1
%o A112519     else: return (k/n)*sum( (-1)^j*binomial(2*n-k-j-1, n-k-j)*binomial(2*j+k-1, j) for j in (0..n-k) )
%o A112519 flatten([[A112519(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jan 12 2022
%Y A112519 Cf. A000108, A055392, A112517, A112520.
%K A112519 easy,nonn,tabl
%O A112519 0,8
%A A112519 _Paul Barry_, Sep 09 2005