A116384 Diagonal sums of the Riordan array A116382.

1, 0, 3, 1, 10, 6, 36, 28, 135, 121, 517, 507, 2003, 2093, 7815, 8569, 30634, 34902, 120480, 141664, 475002, 573574, 1876294, 2318010, 7422676, 9354540, 29400192, 37708672, 116567356, 151868100, 462561572, 611180252, 1836843591, 2458123705

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200

Programs

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

Magma

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]]) >;
[(&+[T(n-k,k): k in [0..Floor(n/2)]]): n in [0..40]];

Mathematica

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 *)

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

Sage

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))
[ sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..40)] # G. C. Greubel, May 22 2019

Formula

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).