A330140 Triangle read by rows: binomial transform of a signed variant of triangle A026300 with alternating signs in each column.

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 1, 4, 9, 0, 0, 1, 5, 15, 21, 0, 0, 1, 6, 24, 50, 51, 0, 0, 1, 7, 35, 98, 161, 127, 0, 0, 1, 8, 48, 168, 378, 504, 323, 0, 0, 1, 9, 63, 264, 750, 1386, 1554, 835, 0, 0, 1, 10, 80, 390, 1335, 3132, 4920, 4740, 2188

Offset: 0

Author

Gary W. Adamson, Dec 02 2019

Keywords

• nonn
• tabl

Examples

			The signed variant of triangle A026300 begins:
   1;
  -1,   1;
   1,  -2,   2;
  -1,   3,  -5,   4;
   1,  -4,   9, -12,   9;
  ...
The binomial transform of the foregoing is as shown below.
Written as a triangle the sequence begins:
  1;
  0,  1;
  0,  0,  2;
  0,  0,  1,  4;
  0,  0,  1,  4,  9;
  0,  0,  1,  5, 15, 21;
  0,  0,  1,  6, 24, 50, 51;
  ...
		

Crossrefs

Right border gives A001006, the Motzkin numbers.

Row sums give A000108, the Catalan numbers.

Cf. A026300, A007318, A330141, A329959.

Programs

Magma

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

Mathematica

F[n_, k_]:= (-1)^(n+k)*Sum[Binomial[n, 2i+n-k]*(Binomial[2i+n-k, i] - Binomial[2i+n-k, i-1]), {i,0,Floor[k/2]}]; T[n_, k_]:= Sum[Binomial[n, j]*F[j, k], {j,k,n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 06 2020 *)