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

Author

Gus Wiseman, Dec 18 2016

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..8000
• Gus Wiseman, Sequences enumerating triangles of integer partitions

Crossrefs

Cf. A000005, A000009, A063834.

Programs

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]

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