%I #10 Sep 08 2022 08:45:24
%S 1,0,3,1,10,6,36,28,135,121,517,507,2003,2093,7815,8569,30634,34902,
%T 120480,141664,475002,573574,1876294,2318010,7422676,9354540,29400192,
%U 37708672,116567356,151868100,462561572,611180252,1836843591,2458123705
%N Diagonal sums of the Riordan array A116382.
%H G. C. Greubel, <a href="/A116384/b116384.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} (-1)^(n-k-j)*C(n-k,j) * Sum_{i=0..j} C(j,i-k)C(i,j-i).
%t T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[n, j]*Sum[Binomial[j, i-k]* Binomial[i, j-i], {i, 0, j}], {j, 0, n}]; Table[Sum[T[n-k, k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* _G. C. Greubel_, May 22 2019 *)
%o (PARI) {T(n,k) = sum(j=0,n, (-1)^(n-j)*binomial(n,j)*sum(m=0,j, binomial(j,m-k)*binomial(m,j-m) ))};vector(40, n, n--; sum(k=0, floor(n/2), T(n-k,k)) ) \\ _G. C. Greubel_, May 22 2019
%o (Magma)
%o T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >;
%o [(&+[T(n-k,k): k in [0..Floor(n/2)]]): n in [0..40]];
%o (Sage)
%o def T(n, k): return sum((-1)^(n-j)*binomial(n,j)*sum(binomial(j,m-k)*binomial(m,j-m) for m in (0..j)) for j in (0..n))
%o [ sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..40)] # _G. C. Greubel_, May 22 2019
%o (GAP) List([0..40], n-> Sum([0..n], k-> Sum([0..n-k], j-> (-1)^(n-k-j)*Binomial(n-k,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m) )))) # _G. C. Greubel_, May 22 2019
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 12 2006