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

0,6

Links

Table of n, a(n) for n=0..65.

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

Right border gives A001006, the Motzkin numbers.

Row sums give A000108, the Catalan numbers.

Cf. A026300, A007318, A330141, A329959.

Sequence in context: A342243 A073429 A123634 * A091866 A168511 A111146

Adjacent sequences: A330137 A330138 A330139 * A330141 A330142 A330143

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 02 2019

Status

approved