A361731 Array read by descending antidiagonals. A(n, k) = hypergeom([-k, -3], [1], n).

1, 1, 1, 1, 4, 1, 1, 10, 7, 1, 1, 20, 25, 10, 1, 1, 35, 63, 46, 13, 1, 1, 56, 129, 136, 73, 16, 1, 1, 84, 231, 307, 245, 106, 19, 1, 1, 120, 377, 586, 593, 396, 145, 22, 1, 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1, 1, 220, 833, 1576, 2073, 2076, 1585, 848, 241, 28, 1

Offset: 0

Author

Peter Luschny, Mar 22 2023

Keywords

• nonn
• tabl

Examples

			Array A(n, k) starts:
 [0] 1,  1,   1,   1,    1,    1,    1,     1, ...  A000012
 [1] 1,  4,  10,  20,   35,   56,   84,   120, ...  A000292
 [2] 1,  7,  25,  63,  129,  231,  377,   575, ...  A001845
 [3] 1, 10,  46, 136,  307,  586, 1000,  1576, ...  A081583
 [4] 1, 13,  73, 245,  593, 1181, 2073,  3333, ...  A081586
 [5] 1, 16, 106, 396, 1011, 2076, 3716,  6056, ...  A081588
 [6] 1, 19, 145, 595, 1585, 3331, 6049,  9955, ...  A081590
 [7] 1, 22, 190, 848, 2339, 5006, 9192, 15240, ...
.
Table T(n, k) starts:
 [0] 1;
 [1] 1,   1;
 [2] 1,   4,   1;
 [3] 1,  10,   7,    1;
 [4] 1,  20,  25,   10,    1;
 [5] 1,  35,  63,   46,   13,    1;
 [6] 1,  56, 129,  136,   73,   16,   1;
 [7] 1,  84, 231,  307,  245,  106,  19,   1;
 [8] 1, 120, 377,  586,  593,  396, 145,  22,  1;
 [9] 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1;
		

Crossrefs

Rows: A000012, A000292, A001845, A081583, A081586, A081588, A081590.

Columns: A000012, A016777, A100536.

Hypergeometric family: A000012 (m=0), A077028 (m=1), A361682 (m=2), this array (m=3).

Programs

Maple

A := (n, k) -> 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6;
seq(print(seq(A(n, k), k = 0..7)), n = 0..7);
# Alternative:
ogf := n -> (1 + (n - 1) * x)^3 / (1 - x)^4:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..9);

Formula

A(n, k) = [x^k] (1 + (n - 1) * x)^3 / (1 - x)^4.

A(n, k) = 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6.

T(n, k) = 1 + (((k*(n - k) - 3*k + 9)*k*(n - k) + (2*k - 9)*k + 18)*k*(n - k))/6.