A279786 Twice-partitioned numbers where the first partition is strict and the latter partitions are constant.

1, 1, 2, 4, 5, 9, 16, 22, 28, 49, 69, 94, 138, 187, 257, 374, 479, 639, 886, 1146, 1577, 2103, 2676, 3534, 4620, 5910, 7542, 9816, 12650, 15986, 20538, 25740, 32632, 41442, 51383, 64771, 81281, 100729, 125041, 155557, 192641, 236810, 293593, 359880, 441276

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..8000

Gus Wiseman, Sequences enumerating triangles of integer partitions

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*d(j)^k*x^(j*k)/k), where d(j) is the number of the divisors of j (A000005). - Ilya Gutkovskiy, Jul 17 2018

Example

The a(5)=9 twice-partitions are:
((5)), ((4)(1)), ((3)(2)), ((3)(11)), ((22)(1)),
((111)(2)), ((111)(11)), ((1111)(1)), ((11111)).

Maple

b:= proc(n, i) option remember;
      `if`(n>i*(i+1)/2, 0, `if`(n=0, 1, b(n, i-1)+
      `if`(i>n, 0, numtheory[tau](i)*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

Mathematica

nn=20; CoefficientList[Series[Product[(1+DivisorSigma[0, n]x^n), {n, nn}], {x, 0, nn}], x]

Crossrefs

Cf. A000005, A000009, A063834.

Sequence in context: A255213 A226447 A073151 * A255515 A067298 A077389

Adjacent sequences: A279783 A279784 A279785 * A279787 A279788 A279789

Keyword

nonn

Author

Gus Wiseman, Dec 18 2016

Status

approved