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

0,3

Links

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

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).

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 *)

Prog

(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
(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]];
(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
(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

Crossrefs

Sequence in context: A107056 A365962 A337273 * A117207 A046658 A124574

Adjacent sequences: A116381 A116382 A116383 * A116385 A116386 A116387

Keyword

easy,nonn

Author

Paul Barry, Feb 12 2006

Status

approved