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A335333
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*(2*k+1)*x + x^2).
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2
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1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 37, 63, 1, 1, 9, 73, 305, 321, 1, 1, 11, 121, 847, 2641, 1683, 1, 1, 13, 181, 1809, 10321, 23525, 8989, 1, 1, 15, 253, 3311, 28401, 129367, 213445, 48639, 1, 1, 17, 337, 5473, 63601, 458649, 1651609, 1961825, 265729, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n,k) is the coefficient of x^n in the expansion of (1 + (2*k+1)*x + k*(k+1)*x^2)^n.
T(n,k) = Sum_{j=0..n} k^j * (k+1)^(n-j) * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (n-1) * T(n-2,k).
T(n,k) = P_n(2*k+1), where P_n is n-th Legendre polynomial.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, ...
1, 13, 37, 73, 121, 181, ...
1, 63, 305, 847, 1809, 3311, ...
1, 321, 2641, 10321, 28401, 63601, ...
1, 1683, 23525, 129367, 458649, 1256651, ...
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MATHEMATICA
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T[n_, k_] := LegendreP[n, 2*k + 1]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)
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PROG
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(PARI) {T(n, k) = pollegendre(n, 2*k+1)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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