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A307884 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2). 4
1, 1, 1, 1, 0, 1, 1, -1, -2, 1, 1, -2, -3, 0, 1, 1, -3, -2, 11, 6, 1, 1, -4, 1, 28, 1, 0, 1, 1, -5, 6, 45, -74, -81, -20, 1, 1, -6, 13, 56, -255, -92, 141, 0, 1, 1, -7, 22, 55, -554, 477, 1324, 363, 70, 1, 1, -8, 33, 36, -959, 2376, 2689, -3656, -1791, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

FORMULA

A(n,k) is the coefficient of x^n in the expansion of (1 - (k-1)*x - k*x^2)^n.

A(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^2.

A(n,k) = Sum_{j=0..n} (-k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).

n * A(n,k) = -(k-1) * (2*n-1) * A(n-1,k) - (k+1)^2 * (n-1) * A(n-2,k).

EXAMPLE

Square array begins:

   1,   1,   1,    1,    1,    1,      1, ...

   1,   0,  -1,   -2,   -3,   -4,     -5, ...

   1,  -2,  -3,   -2,    1,    6,     13, ...

   1,   0,  11,   28,   45,   56,     55, ...

   1,   6,   1,  -74, -255, -554,   -959, ...

   1,   0, -81,  -92,  477, 2376,   6475, ...

   1, -20, 141, 1324, 2689, -804, -20195, ...

CROSSREFS

Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341,

Main diagonal gives A307885.

Cf. A307883.

Sequence in context: A214751 A306512 A239397 * A329221 A177858 A166967

Adjacent sequences:  A307881 A307882 A307883 * A307885 A307886 A307887

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, May 02 2019

STATUS

approved

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Last modified February 19 19:28 EST 2020. Contains 332047 sequences. (Running on oeis4.)