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A361731
Array read by descending antidiagonals. A(n, k) = hypergeom([-k, -3], [1], n).
1
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
0,5
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.
EXAMPLE
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;
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);
CROSSREFS
Columns: A000012, A016777, A100536.
Hypergeometric family: A000012 (m=0), A077028 (m=1), A361682 (m=2), this array (m=3).
Sequence in context: A369189 A185027 A016520 * A109955 A214398 A220860
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 22 2023
STATUS
approved