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

Author

Paul Barry, Apr 15 2010

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A176331, A176332, A176335.

Programs

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

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

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 *)

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

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

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).

a(n) ~ phi^(2*n+3) / (4*5^(1/4)*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 08 2024