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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

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, ...

CROSSREFS

Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.

T(n,n+1) gives A331794.

Cf. A307883, A331511, A331514, A331795.

Sequence in context: A340313 A120873 A125161 * A125933 A255054 A011857

Adjacent sequences:  A331788 A331789 A331790 * A331792 A331793 A331794

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Jan 26 2020

STATUS

approved

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Last modified April 15 02:27 EDT 2021. Contains 342974 sequences. (Running on oeis4.)