A350584 Triangle read by rows, T(n, k) = [x^k] ((2*x^3 - 3*x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.

1, 1, 3, 1, 4, 7, 1, 5, 12, 19, 1, 6, 18, 37, 56, 1, 7, 25, 62, 118, 174, 1, 8, 33, 95, 213, 387, 561, 1, 9, 42, 137, 350, 737, 1298, 1859, 1, 10, 52, 189, 539, 1276, 2574, 4433, 6292, 1, 11, 63, 252, 791, 2067, 4641, 9074, 15366, 21658

Offset: 1

Author

Peter Luschny, Mar 27 2022

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[1] [1]
[2] [1,  3]
[3] [1,  4,  7]
[4] [1,  5, 12,  19]
[5] [1,  6, 18,  37,  56]
[6] [1,  7, 25,  62, 118,  174]
[7] [1,  8, 33,  95, 213,  387,  561]
[8] [1,  9, 42, 137, 350,  737, 1298, 1859]
[9] [1, 10, 52, 189, 539, 1276, 2574, 4433, 6292]
		

Crossrefs

A280891 (row sums), A135339 (alternating row sums), A005807 or A071716 (main diagonal).

Programs

Maple

# Compare the analogue algorithm for the Bell triangle in A046937.
A350584Triangle := proc(len) local A, P, T, n; A := [2]; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([op(P), A[-1]]);
A := P; T := [op(T), P] od; T end:
A350584Triangle(10): ListTools:-Flatten(%);
# Alternative:
ogf := n -> (2*x^3 - 3*x^2 - x + 1)/(1 - x)^(n + 2):
ser := n -> series(ogf(n), x, n):
row := n -> seq(coeff(ser(n), x, k), k = 0..n-1):
seq(row(n), n = 1..10);