A047913 Triangle of numbers a(n,k) = number of partitions of k such that k = n + n_1 + n_2 + ... + n_t where n_1 <= 2n and n_{i+1} <= 2n_i for all i.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 7, 9, 1, 1, 2, 4, 7, 12, 16, 1, 1, 2, 4, 7, 13, 22, 28, 1, 1, 2, 4, 7, 13, 24, 39, 50, 1, 1, 2, 4, 7, 13, 24, 42, 70, 89, 1, 1, 2, 4, 7, 13, 24, 43, 76, 126, 159, 1, 1, 2, 4, 7, 13, 24, 43, 78, 137, 225, 285

Offset: 1

Author

N. J. A. Sloane

Keywords

• tabl
• nonn
• easy
• nice

Comments

Triangle is read in this order: a(1,1), a(2,2), a(1,2), a(3,3), a(2,3), a(1,3), a(4,4), ...

Rows are the columns in the table at the end of the Minc reference, read bottom to top. - Joerg Arndt, Jan 15 2024

Examples

			Triangle begins:
1;
1, 1;
1, 1, 2;
1, 1, 2, 3;
1, 1, 2, 4, 5;
1, 1, 2, 4, 7,  9;
1, 1, 2, 4, 7, 12, 16;
1, 1, 2, 4, 7, 13, 22, 28;
1, 1, 2, 4, 7, 13, 24, 39, 50;
1, 1, 2, 4, 7, 13, 24, 42, 70,  89;
1, 1, 2, 4, 7, 13, 24, 43, 76, 126, 159;
1, 1, 2, 4, 7, 13, 24, 43, 78, 137, 225, 285;
...
Rows approach A002843. - _Joerg Arndt_, Jan 15 2024
		

Links

• H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

Crossrefs

Rows give A002572, A002573, A002574, ..., columns approach A002843.

Cf. A049286 (triangle with reversed rows).

Programs

Mathematica

a[n_, n_] = 1; a[n_?Positive, k_?Positive] := a[n, k] = Sum[a[i, k-n], {i, 1, 2*n}]; a[n_, k_] = 0; Table[a[n, k], {k, 1, 12}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Oct 21 2013 *)

Formula

a(n, n)=1, a(n, k) = Sum_{i=1..2n} a(i, k-n).