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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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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
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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
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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 *) -
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
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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
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).
Comments