A176334 Diagonal sums of number triangle A176331.

1, 1, 2, 4, 9, 21, 51, 124, 305, 755, 1879, 4698, 11792, 29694, 74984, 189811, 481498, 1223713, 3115200, 7942134, 20275362, 51823246, 132604193, 339644739, 870745187, 2234208932, 5737129623, 14742751524, 37909928908, 97543380598

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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

Maple

A176334 := proc(n)
    add(add(binomial(j, n-2*k)*binomial(j, k)*(-1)^(n-k-j), j=0..n-k), k=0..floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015

Mathematica

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j, 0, n}]; Table[Sum[T[n-k, k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Dec 07 2019 *)

Prog

(PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(30, n, sum(j=0, (n-1)\2, T(n-j-1, j)) ) \\ G. C. Greubel, Dec 07 2019
(Magma) T:= func< n, k | &+[(-1)^(n-j)*Binomial(j, n-k)*Binomial(j, k): j in [0..n]] >;
[(&+[T(n-k, k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Dec 07 2019
(Sage)
@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Dec 07 2019
(GAP)
T:= function(n, k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j, k)*Binomial(j, n-k) );
  end;
List([0..30], n-> Sum([0..Int(n/2)], j-> T(n-j, j) )); # G. C. Greubel, Dec 07 2019

Crossrefs

Cf. A176331, A176332, A176335.

Sequence in context: A199410 A091600 A261232 * A257386 A048285 A051529

Adjacent sequences: A176331 A176332 A176333 * A176335 A176336 A176337

Keyword

easy,nonn

Author

Paul Barry, Apr 15 2010

Status

approved