A144399 Triangle in A144385 with rows left-adjusted.

1, 1, 1, 1, 1, 3, 7, 10, 10, 1, 6, 25, 75, 175, 280, 280, 1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400, 1, 15, 140, 980, 5565, 26145, 102025, 323400, 800800, 1401400, 1401400, 1, 21, 266, 2520, 19425, 125895, 695695, 3273270, 12962950

Offset: 0

Author

David Applegate and N. J. A. Sloane, Dec 07 2008

Keywords

• nonn
• tabf

Comments

Row n has 2n+1 terms.

Examples

			Triangle begins:
1
1, 1, 1
1, 3, 7, 10, 10
1, 6, 25, 75, 175, 280, 280
1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Crossrefs

Cf. A144385. Row sums give A144416.

Programs

Maple

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
       b(n-j)*binomial(n-1, j-1), j=1..min(3, n))*x))
    end:
T:= (n, k)-> coeff(b(k), x, n):
seq(seq(T(n, k), k=n..3*n), n=0..6);  # Alois P. Heinz, May 31 2018

Mathematica

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - j]*Binomial[n - 1, j - 1], {j, 1, Min[3, n]}]*x]];
T[n_, k_] := Coefficient[b[k], x, n];
Table[T[n, k], {n, 0, 6}, { k, n, 3n}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)