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A116384
Diagonal sums of the Riordan array A116382.
2
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
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: A365962 A337273 A376787 * A117207 A046658 A124574
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 12 2006
STATUS
approved