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

0,3

Comments

Row sums = A007317(n+1).

Right border = A000108(n+1).

Links

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

Formula

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

Example

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;
  ...

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A000108, A007317, A007318, A050166.

Sequence in context: A302689 A289088 A057611 * A259701 A147843 A094597

Adjacent sequences: A329956 A329957 A329958 * A329960 A329961 A329962

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 25 2019

Extensions

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

Status

approved