|
%I #8 Mar 23 2023 07:57:42
%S 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,
%T 1,1,84,231,307,245,106,19,1,1,120,377,586,593,396,145,22,1,1,165,575,
%U 1000,1181,1011,595,190,25,1,1,220,833,1576,2073,2076,1585,848,241,28,1
%N Array read by descending antidiagonals. A(n, k) = hypergeom([-k, -3], [1], n).
%F A(n, k) = [x^k] (1 + (n - 1) * x)^3 / (1 - x)^4.
%F A(n, k) = 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6.
%F T(n, k) = 1 + (((k*(n - k) - 3*k + 9)*k*(n - k) + (2*k - 9)*k + 18)*k*(n - k))/6.
%e Array A(n, k) starts:
%e [0] 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%e [1] 1, 4, 10, 20, 35, 56, 84, 120, ... A000292
%e [2] 1, 7, 25, 63, 129, 231, 377, 575, ... A001845
%e [3] 1, 10, 46, 136, 307, 586, 1000, 1576, ... A081583
%e [4] 1, 13, 73, 245, 593, 1181, 2073, 3333, ... A081586
%e [5] 1, 16, 106, 396, 1011, 2076, 3716, 6056, ... A081588
%e [6] 1, 19, 145, 595, 1585, 3331, 6049, 9955, ... A081590
%e [7] 1, 22, 190, 848, 2339, 5006, 9192, 15240, ...
%e .
%e Table T(n, k) starts:
%e [0] 1;
%e [1] 1, 1;
%e [2] 1, 4, 1;
%e [3] 1, 10, 7, 1;
%e [4] 1, 20, 25, 10, 1;
%e [5] 1, 35, 63, 46, 13, 1;
%e [6] 1, 56, 129, 136, 73, 16, 1;
%e [7] 1, 84, 231, 307, 245, 106, 19, 1;
%e [8] 1, 120, 377, 586, 593, 396, 145, 22, 1;
%e [9] 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1;
%p A := (n, k) -> 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6;
%p seq(print(seq(A(n, k), k = 0..7)), n = 0..7);
%p # Alternative:
%p ogf := n -> (1 + (n - 1) * x)^3 / (1 - x)^4:
%p ser := n -> series(ogf(n), x, 12):
%p row := n -> seq(coeff(ser(n), x, k), k = 0..9):
%p seq(print(row(n)), n = 0..9);
%Y Rows: A000012, A000292, A001845, A081583, A081586, A081588, A081590.
%Y Columns: A000012, A016777, A100536.
%Y Hypergeometric family: A000012 (m=0), A077028 (m=1), A361682 (m=2), this array (m=3).
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Mar 22 2023
|
