A307079 - OEIS

A307079

Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k+x^k).

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 0, 1, 1, 2, 3, 3, -4, 1, 1, 2, 3, 4, 0, -8, 1, 1, 2, 3, 4, 4, -9, -8, 1, 1, 2, 3, 4, 5, 0, -27, 0, 1, 1, 2, 3, 4, 5, 5, -14, -54, 16, 1, 1, 2, 3, 4, 5, 6, 0, -48, -81, 32, 1, 1, 2, 3, 4, 5, 6, 6, -20, -116, -81, 32, 1

(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+1,k*j+1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i,k*j) * binomial(n-i,k*j).

EXAMPLE

Square array begins:

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

1, 2, 2, 2, 2, 2, 2, 2, 2, ...

1, 2, 3, 3, 3, 3, 3, 3, 3, ...

1, 0, 3, 4, 4, 4, 4, 4, 4, ...

1, -4, 0, 4, 5, 5, 5, 5, 5, ...

1, -8, -9, 0, 5, 6, 6, 6, 6, ...

1, -8, -27, -14, 0, 6, 7, 7, 7, ...

1, 0, -54, -48, -20, 0, 7, 8, 8, ...

1, 16, -81, -116, -75, -27, 0, 8, 9, ...

MATHEMATICA

T[n_, k_] := Sum[(-1)^j * Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

CROSSREFS

Columns 1-6 give A000012, A099087, A057682(n+1), A099587(n+1), A289321(n+1), A307089.

Cf. A306914, A307039, A307078.

Sequence in context: A113453 A245851 A230596 * A330190 A356300 A348041

Adjacent sequences: A307076 A307077 A307078 * A307080 A307081 A307082

KEYWORD

sign,tabl,look

AUTHOR

Seiichi Manyama, Mar 22 2019

STATUS

approved