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A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)). 4
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 9, 1, 1, 1, 5, 10, 21, 21, 1, 1, 1, 6, 13, 37, 61, 51, 1, 1, 1, 7, 16, 57, 121, 191, 127, 1, 1, 1, 8, 19, 81, 201, 451, 603, 323, 1, 1, 1, 9, 22, 109, 301, 861, 1639, 1961, 835, 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/(n+1) * (1 + x + k*x^2)^(n+1).

A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).

(n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

EXAMPLE

Square array begins:

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

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

   1,   2,   3,    4,    5,    6,     7,     8, ...

   1,   4,   7,   10,   13,   16,    19,    22, ...

   1,   9,  21,   37,   57,   81,   109,   141, ...

   1,  21,  61,  121,  201,  301,   421,   561, ...

   1,  51, 191,  451,  861, 1451,  2251,  3291, ...

   1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...

CROSSREFS

Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.

Main diagonal gives A307906.

Cf. A000108, A107267, A247495, A307855.

Sequence in context: A173072 A219272 A084097 * A293991 A288638 A261494

Adjacent sequences:  A306681 A306682 A306683 * A306685 A306686 A306687

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, May 06 2019

STATUS

approved

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Last modified May 26 00:32 EDT 2020. Contains 334613 sequences. (Running on oeis4.)