A296180 Triangle read by rows: T(n, k) = 3*(n - k)*k + 1, n >= 0, 0 <= k <= n.

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 13, 10, 1, 1, 13, 19, 19, 13, 1, 1, 16, 25, 28, 25, 16, 1, 1, 19, 31, 37, 37, 31, 19, 1, 1, 22, 37, 46, 49, 46, 37, 22, 1, 1, 25, 43, 55, 61, 61, 55, 43, 25, 1, 1, 28, 49, 64, 73, 76, 73, 64, 49, 28, 1

Offset: 0

Author

Wolfdieter Lang, Dec 20 2017

Keywords

• nonn
• easy
• tabl

Comments

This is member m = 3 of the family of triangles T(m; n, k) = m*(n - k)*k + 1, for m >= 0. For m = 0: A000012(n, k) (read as a triangle); for m = 1: A077028 (rascal), for m = 2: T(2, n+1, k+1) = A130154(n, k). Motivated by A130154 to look at this family of triangles.

In general the recurrence is: T(m; n, 0) = 1 and T(m; n, n) = 1 for n >= 0; T(m; n, k) = (T(m; n-1, k-1)*T(m; n-1, k) + m)/T(m; n-2, k-1), for n >= 2, k = 1..n-1.

The general g.f. of the sequence of column k (with leading zeros) is G(m; k, x) = (x^k/(1 - x)^2)*(1 + (m*k - 1)*x), k >= 0.

The general g.f. of the triangle T(m;, n, k) is GT(m; x, t) = (1 - (1 + t)*x + (m+1)*t*x^2)/((1 - t*x)*(1 - x))^2, and G(m; k, x) = (d/dt)^k GT(m; x, t)/k!|_{t=0}.

For a simple combinatorial interpretation see the one given in A130154 by Rogério Serôdio which can be generalized to m >= 3.

Examples

			The triangle T(n, k) begins:
n\k   0  1  2  3  4  5  6  7  8  9 10 ...
0:    1
1:    1  1
2:    1  4  1
3:    1  7  7  1
4:    1 10 13 10  1
5:    1 13 19 19 13  1
6:    1 16 25 28 25 16  1
7:    1 19 31 37 37 31 19  1
8:    1 22 37 46 49 46 37 22  1
9:    1 25 43 55 61 61 55 43 25  1
10:   1 28 49 64 73 76 73 64 49 28  1
...
Recurrence: 28 = T(6, 3) = (19*19 + 3)/13 = 28.
		

Crossrefs

Cf. A077028, A130154.

Columns (without leading zeros): A000012, A016777, A016921, A016921, A017173, A017533, ...

Programs

Mathematica

Table[3 k (n - k) + 1, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 20 2017 *)

PARI

lista(nn) = for(n=0, nn, for(k=0, n, print1(3*(n - k)*k + 1, ", "))) \\ Iain Fox, Dec 21 2017

Formula

T(n, k) = 3*(n - k)*k + 1, n >= 0, 0 <= k <= n,

Recurrence: T(n, 0) = 1 and T(n, n) = 1 for n >= 0; T(n, k) = (T(n-1, k-1)*T(n-1, k) + 3)/T(n-2, k-1), for n >= 2, k = 1..n-1.

G.f. of column k (with leading zeros): (x^k/(1 - x)^2)*(1 + (3*k-1)*x), k >= 0.

G.f. of triangle: (1 - (1 + t)*x + 4*t*x^2)/((1 - t*x)*(1 - x))^2 = 1 + (1+t)*x +(1 + 4*t + t^2)*x^2 + (1 + 7*t + 7*t^2 + t^3)*x^3 = ...