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A307847
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Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k^2)*x^2).
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5
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1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 7, 1, 1, 1, 19, 25, 19, 1, 1, 1, 33, 55, 145, 51, 1, 1, 1, 51, 97, 595, 561, 141, 1, 1, 1, 73, 151, 1729, 2611, 2841, 393, 1, 1, 1, 99, 217, 4051, 8001, 22141, 12489, 1107, 1, 1, 1, 129, 295, 8209, 19251, 105441, 119449, 60705, 3139, 1
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OFFSET
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0,9
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LINKS
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FORMULA
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A(n,k) is the constant term in the expansion of (k/x + 1 + k*x)^n.
A(n,k) is the coefficient of x^n in the expansion of (1 + x + (k*x)^2)^n.
A(n,k) = Sum_{j=0..n} (1-2*k)^(n-j) * k^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (1+2*k)^(n-j) * (-k)^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k^2) * (n-1) * A(n-2,k).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 3, 9, 19, 33, 51, ...
1, 7, 25, 55, 97, 151, ...
1, 19, 145, 595, 1729, 4051, ...
1, 51, 561, 2611, 8001, 19251, ...
1, 141, 2841, 22141, 105441, 369501, ...
1, 393, 12489, 119449, 627873, 2319801, ...
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MATHEMATICA
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T[n_, k_] := Sum[If[k == j == 0, 1, k^(2*j)] * Binomial[n, 2*j] * Binomial[2*j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)
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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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