A329066 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( (Sum_{j=0..n} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n} y^(2*j+1)+1/y^(2*j+1)) - (Sum_{j=0..n-1} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n-1} y^(2*j+1)+1/y^(2*j+1)) )^(2*k).
1, 4, 1, 36, 12, 1, 400, 588, 20, 1, 4900, 49440, 2100, 28, 1, 63504, 5187980, 423440, 4956, 36, 1, 853776, 597027312, 117234740, 1751680, 9540, 44, 1, 11778624, 71962945824, 36938855520, 907687900, 5101200, 16236, 52, 1
Offset: 0
Examples
Square array begins: 1, 4, 36, 400, 4900, ... 1, 12, 588, 49440, 5187980, ... 1, 20, 2100, 423440, 117234740, ... 1, 28, 4956, 1751680, 907687900, ... 1, 36, 9540, 5101200, 4190017860, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..50, flattened
- Wikipedia, Taxicab geometry.
Crossrefs
Programs
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PARI
{T(n, k) = polcoef(polcoef((sum(j=0, 2*n+1, (x^j+1/x^j)*(y^(2*n+1-j)+1/y^(2*n+1-j)))-x^(2*n+1)-1/x^(2*n+1)-y^(2*n+1)-1/y^(2*n+1))^(2*k), 0), 0)} -
PARI
f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x); T(n, k) = sum(j=0, 2*k, (-1)^j*binomial(2*k, j)*polcoef(f(n)^j*f(n-1)^(2*k-j), 0)^2)
Formula
See the second code written in PARI.
Comments