A145153 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where sequence a_k of column k is the expansion of x/((1 - x - x^4)*(1 - x)^(k - 1)).

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 1, 0, 1, 4, 6, 4, 2, 1, 0, 1, 5, 10, 10, 6, 3, 1, 0, 1, 6, 15, 20, 16, 9, 4, 1, 0, 1, 7, 21, 35, 36, 25, 13, 5, 2, 0, 1, 8, 28, 56, 71, 61, 38, 18, 7, 3, 0, 1, 9, 36, 84, 127, 132, 99, 56, 25, 10, 4, 0, 1, 10, 45, 120, 211, 259, 231, 155, 81, 35, 14, 5

Offset: 0

Author

Alois P. Heinz, Oct 03 2008

Keywords

• nonn
• tabl

Comments

Each row sequence a_n (for n > 0) is produced by a polynomial of degree n-1, whose (rational) coefficients are given in row n of A145140/A145141. The coefficients *(n-1)! are given in A145142.

Each column sequence a_k is produced by a recursion, whose coefficients are given by row k of A145152.

Examples

			Square array A(n,k) begins:
  0, 0, 0,  0,  0,  0,   0, ...
  1, 1, 1,  1,  1,  1,   1, ...
  0, 1, 2,  3,  4,  5,   6, ...
  0, 1, 3,  6, 10, 15,  21, ...
  0, 1, 4, 10, 20, 35,  56, ...
  1, 2, 6, 16, 36, 71, 127, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Rows 0-9 give: A000004, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130.

Columns 0-9 give: A017898(n-1) for n>0, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.

Main diagonal gives: A145138.

Antidiaginal sums give: A145139.

Numerators/denominators of polynomials for rows give: A145140/A145141.

Cf. A145142, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150, A145151, A145152.

Programs

Maple

A:= proc(n, k) coeftayl (x/ (1-x-x^4)/ (1-x)^(k-1), x=0, n) end:
seq(seq(A(n, d-n), n=0..d), d=0..13);

Mathematica

a[n_, k_] := SeriesCoefficient[x/(1 - x - x^4)/(1 - x)^(k - 1), {x, 0, n}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 05 2013 *)

Formula

G.f. of column k: x/((1-x-x^4)*(1-x)^(k-1)).