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A331791
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2*k*x + ((k-2)*x)^2 + (1 - k*x) * sqrt(1 - 2*k*x + ((k-2)*x)^2)).
5
1, 1, 0, 1, 2, -3, 1, 4, 3, 0, 1, 6, 15, 4, 10, 1, 8, 33, 56, 5, 0, 1, 10, 57, 180, 210, 6, -35, 1, 12, 87, 400, 985, 792, 7, 0, 1, 14, 123, 740, 2810, 5418, 3003, 8, 126, 1, 16, 165, 1224, 6285, 19824, 29953, 11440, 9, 0
OFFSET
0,5
LINKS
FORMULA
T(n,k) = Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
n * (n+2) * T(n,k) = (n+1) * (k * (2*n+1) * T(n-1,k) - (k-2)^2 * n * T(n-2,k)) for n>1.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
-3, 3, 15, 33, 57, 87, ...
0, 4, 56, 180, 400, 740, ...
10, 5, 210, 985, 2810, 6285, ...
0, 6, 792, 5418, 19824, 53550, ...
MATHEMATICA
T[n_, k_] := Sum[If[k==1 && j==0, 1, (k-1)^j] * Binomial[n + 1, j] * Binomial[n + 1, j + 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 2021 *)
CROSSREFS
Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.
T(n,n+1) gives A331794.
Sequence in context: A340313 A120873 A125161 * A365327 A125933 A255054
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jan 26 2020
STATUS
approved