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

%I A361731 #8 Mar 23 2023 07:57:42
%S A361731 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 A361731 1,1,84,231,307,245,106,19,1,1,120,377,586,593,396,145,22,1,1,165,575,
%U A361731 1000,1181,1011,595,190,25,1,1,220,833,1576,2073,2076,1585,848,241,28,1
%N A361731 Array read by descending antidiagonals. A(n, k) = hypergeom([-k, -3], [1], n).
%F A361731 A(n, k) = [x^k] (1 + (n - 1) * x)^3 / (1 - x)^4.
%F A361731 A(n, k) = 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6.
%F A361731 T(n, k) = 1 + (((k*(n - k) - 3*k + 9)*k*(n - k) + (2*k - 9)*k + 18)*k*(n - k))/6.
%e A361731 Array A(n, k) starts:
%e A361731  [0] 1,  1,   1,   1,    1,    1,    1,     1, ...  A000012
%e A361731  [1] 1,  4,  10,  20,   35,   56,   84,   120, ...  A000292
%e A361731  [2] 1,  7,  25,  63,  129,  231,  377,   575, ...  A001845
%e A361731  [3] 1, 10,  46, 136,  307,  586, 1000,  1576, ...  A081583
%e A361731  [4] 1, 13,  73, 245,  593, 1181, 2073,  3333, ...  A081586
%e A361731  [5] 1, 16, 106, 396, 1011, 2076, 3716,  6056, ...  A081588
%e A361731  [6] 1, 19, 145, 595, 1585, 3331, 6049,  9955, ...  A081590
%e A361731  [7] 1, 22, 190, 848, 2339, 5006, 9192, 15240, ...
%e A361731 .
%e A361731 Table T(n, k) starts:
%e A361731  [0] 1;
%e A361731  [1] 1,   1;
%e A361731  [2] 1,   4,   1;
%e A361731  [3] 1,  10,   7,    1;
%e A361731  [4] 1,  20,  25,   10,    1;
%e A361731  [5] 1,  35,  63,   46,   13,    1;
%e A361731  [6] 1,  56, 129,  136,   73,   16,   1;
%e A361731  [7] 1,  84, 231,  307,  245,  106,  19,   1;
%e A361731  [8] 1, 120, 377,  586,  593,  396, 145,  22,  1;
%e A361731  [9] 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1;
%p A361731 A := (n, k) -> 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6;
%p A361731 seq(print(seq(A(n, k), k = 0..7)), n = 0..7);
%p A361731 # Alternative:
%p A361731 ogf := n -> (1 + (n - 1) * x)^3 / (1 - x)^4:
%p A361731 ser := n -> series(ogf(n), x, 12):
%p A361731 row := n -> seq(coeff(ser(n), x, k), k = 0..9):
%p A361731 seq(print(row(n)), n = 0..9);
%Y A361731 Rows: A000012, A000292, A001845, A081583, A081586, A081588, A081590.
%Y A361731 Columns: A000012, A016777, A100536.
%Y A361731 Hypergeometric family: A000012 (m=0), A077028 (m=1), A361682 (m=2), this array (m=3).
%K A361731 nonn,tabl
%O A361731 0,5
%A A361731 _Peter Luschny_, Mar 22 2023