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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 (list; graph; refs; listen; history; text; internal format)
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: A091965 A171568 A107056 * A117207 A046658 A124574

Adjacent sequences:  A116381 A116382 A116383 * A116385 A116386 A116387

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 12 2006

STATUS

approved

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Last modified November 17 03:06 EST 2019. Contains 329216 sequences. (Running on oeis4.)