A329959 Binomial transform of a signed variant of triangle A050166.

1, 0, 2, 0, 0, 5, 0, 0, 1, 14, 0, 0, 1, 8, 42, 0, 0, 1, 10, 45, 132, 0, 0, 1, 12, 69, 220, 429, 0, 0, 1, 14, 98, 406, 1001, 1430, 0, 0, 1, 16, 132, 672, 2184, 4368, 4862, 0, 0, 1, 18, 171, 1032, 4152, 11088, 18564, 16796, 0, 0, 1, 20, 215, 1500, 7185, 23904, 54060, 77520, 58786

Offset: 0

Author

Gary W. Adamson, Nov 25 2019

Keywords

• nonn
• tabl

Comments

Row sums = A007317(n+1).

Right border = A000108(n+1).

Examples

			The signed variant of A050166 is A050166(n,k) * (-1)^(n+k):
   1;
  -1,   2;
   1,  -4,   5;
  -1,   6, -14,  14;
   1,  -8,  27, -48,  42;
  ...
Let the above triangle = S, and Pascal's triangle = P as an infinite lower triangular matrix. Then T = P * S gives:
  1;
  0,   2;
  0,   0,   5;
  0,   0,   1,  14;
  0,   0,   1,   8,  42;
  0,   0,   1,  10,  45, 132;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A000108, A007317, A007318, A050166.

Programs

GAP

B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> Sum([k..n], j-> (-1)^(k+j)*B(n,j)*(B(2*j,k) - B(2*j,k-2)) )))); # G. C. Greubel, Jan 06 2020

Magma

T:= func< n,k | &+[(-1)^(k+j)*Binomial(n,j)*(Binomial(2*j,k) - Binomial(2*j,k-2)): j in [k..n]] >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 06 2020

Maple

S:= (n, k)-> (binomial(2*n, k)-binomial(2*n, k-2))*(-1)^(n+k):
T:= (n, k)-> add(binomial(n, j)*S(j, k), j=k..n):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Nov 27 2019

Mathematica

Table[Sum[(-1)^(k+j)*Binomial[n, j]*(Binomial[2*j, k] - Binomial[2*j, k-2]), {j, k, n}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 06 2020 *)

PARI

T(n,k) = sum(j=k, n, (-1)^(k+j)*binomial(n,j)*(binomial(2*j,k) - binomial(2*j,k-2)) ); \\ G. C. Greubel, Jan 06 2020

Sage

[[sum((-1)^(k+j)*binomial(n,j)*(binomial(2*j,k) - binomial(2*j,k-2)) for j in (k..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 06 2020

Formula

T(n,k) = Sum_{j=k..n} C(n,j) * (-1)^(j+k) * A050166(j,k). - Alois P. Heinz, Nov 27 2019

Extensions

New offset and more terms from Alois P. Heinz, Nov 25 2019