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

1,6

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

Links

Table of n, a(n) for n=1..78.

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.

Formula

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

Example

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

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 *)

Crossrefs

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

Cf. A049286 (triangle with reversed rows).

Sequence in context: A360492 A360491 A360333 * A152977 A360334 A259799

Adjacent sequences: A047910 A047911 A047912 * A047914 A047915 A047916

Keyword

tabl,nonn,easy,nice

Author

N. J. A. Sloane

Status

approved