(MAGMAMagma)

reviewed

approved

proposed

reviewed

editing

proposed

(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

proposed

editing

editing

proposed

G. C. Greubel, <a href="/A116384/b116384.txt">Table of n, a(n) for n = 0..200</a>

a(n) =sum Sum_{k=0..floor(n/2), sum} Sum_{j=0..n-k, } (-1)^(n-k-j)*C(n-k,j) *sum Sum_{i=0..j, } C(j,i-k)C(i,j-i)}}}.

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

(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

approved

editing

_Paul Barry (pbarry(AT)wit.ie), _, Feb 12 2006

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

0,3

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)}}}.

easy,nonn

Paul Barry (pbarry(AT)wit.ie), Feb 12 2006

approved