A123878 Product of signed and unsigned Morgan-Voyce triangles.

1, 0, 1, -1, 0, 1, -1, -3, 0, 1, 0, -3, -5, 0, 1, 1, 3, -5, -7, 0, 1, 1, 9, 10, -7, -9, 0, 1, 0, 5, 25, 21, -9, -11, 0, 1, -1, -9, 5, 49, 36, -11, -13, 0, 1, -1, -18, -50, -7, 81, 55, -13, -15, 0, 1, 0, -7, -70, -147, -39, 121, 78, -15, -17, 0, 1

Offset

0,8

Comments

Inverse is A123880.

Row sums are A123879.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

Riordan array ((1-x)/(1-x+x^2), x*(1-x)^2/(1-x+x^2)^2).

Number triangle: T(n,k) = Sum_{j=0..n} C(n+j,2*j)*C(j+k,2*k)*(-1)^(j-k).

Example

Number triangle begins:
   1;
   0,  1;
  -1,  0,  1;
  -1, -3,  0,  1;
   0, -3, -5,  0,  1;
   1,  3, -5, -7,  0, 1;
   1,  9, 10, -7, -9, 0, 1;

Mathematica

Table[Sum[(-1)^(j-k)*Binomial[n+j, 2*j]*Binomial[j+k, 2*k], {j, 0, n}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 08 2019 *)

Prog

(PARI) T(n, k) = sum(j=0, n, (-1)^(j-k)*binomial(n+j, 2*j)*binomial(n+j, 2*k) );
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 08 2019
(Magma) B:= Binomial; [(&+[(-1)^(j-k)*B(n+j, 2*j)*B(n+j, 2*k):j in [0..n]]) : k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 08 2019
(Sage) b=binomial; [[sum((-1)^(j-k)*b(n+j, 2*j)*b(n+j, 2*k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 08 2019
(GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(j-k)*B(n+j, 2*j)*B(n+j, 2*k) )))); # G. C. Greubel, Aug 08 2019

Crossrefs

Cf. A085478, A123879, A123880.

Sequence in context: A340504 A242447 A238342 * A284148 A108197 A318455

Adjacent sequences: A123875 A123876 A123877 * A123879 A123880 A123881

Keyword

easy,sign,tabl

Author

Paul Barry, Oct 16 2006

Status

approved