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A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k). 11
1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 0, 0, 1, 5, 10, 9, -4, 0, 1, 6, 15, 20, 9, -8, 0, 1, 7, 21, 35, 34, 0, -8, 0, 1, 8, 28, 56, 70, 48, -27, 0, 0, 1, 9, 36, 84, 126, 125, 48, -81, 16, 0, 1, 10, 45, 120, 210, 252, 200, 0, -162, 32, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

FORMULA

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

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

EXAMPLE

Square array begins:

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

   0,  2,    3,    4,   5,    6,    7,    8, ...

   0,  2,    6,   10,  15,   21,   28,   36, ...

   0,  0,    9,   20,  35,   56,   84,  120, ...

   0, -4,    9,   34,  70,  126,  210,  330, ...

   0, -8,    0,   48, 125,  252,  462,  792, ...

   0, -8,  -27,   48, 200,  461,  924, 1716, ...

   0,  0,  -81,    0, 275,  780, 1715, 3432, ...

   0, 16, -162, -164, 275, 1209, 2989, 6434, ...

MATHEMATICA

A[n_, k_] := SeriesCoefficient[1/((1-x)^k + x^k), {x, 0, n}];

Table[A[n-k+1, k], {n, 0, 11}, {k, n+1, 1, -1}] // Flatten (* Jean-Fran├žois Alcover, Mar 20 2019 *)

CROSSREFS

Columns 1-9 give A000007, A099087, A057083, A099589(n+3), A289389(n+4), A306940, (-1)^n * A049018(n), A306941, A306942.

Cf. A039912, A306913, A306915, A307039, A307079, A307394.

Sequence in context: A253668 A216220 A216235 * A317023 A319284 A182703

Adjacent sequences:  A306911 A306912 A306913 * A306915 A306916 A306917

KEYWORD

sign,tabl,look

AUTHOR

Seiichi Manyama, Mar 16 2019

STATUS

approved

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Last modified May 27 02:54 EDT 2019. Contains 323597 sequences. (Running on oeis4.)