A306941 -id:A306941

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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).

+10
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-François 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.

KEYWORD

sign,tabl,look

AUTHOR

Seiichi Manyama, Mar 16 2019

STATUS

approved

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