A279784 Twice partitioned numbers where the latter partitions are constant.

1, 1, 3, 5, 12, 18, 40, 60, 121, 186, 344, 524, 955, 1432, 2484, 3756, 6352, 9493, 15750, 23414, 38128, 56513, 90406, 133312, 211194, 309657, 484214, 708267, 1097159, 1597290, 2454245, 3560444, 5430091, 7854174, 11894335, 17151394, 25838413, 37145198, 55648059

Offset: 0

Author

Gus Wiseman, Dec 18 2016

Keywords

• nonn

Comments

Also number of ways to choose a divisor of each part of an integer partition of n.

Examples

			The a(4)=12 twice-partitions are:
((4)), ((3)(1)), ((2)(2)), ((22)),
((2)(1)(1)), ((2)(11)), ((11)(2)),
((1)(1)(1)(1)), ((11)(1)(1)), ((11)(11)), ((111)(1)), ((1111)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• Gus Wiseman, Sequences enumerating triangles of integer partitions

Crossrefs

Cf. A000005, A063834, A145515, A270995.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+`if`(i>n, 0, numtheory[tau](i)*b(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 20 2016

Mathematica

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

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=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

From Vaclav Kotesovec, Jul 28 2018: (Start)

a(n) ~ c * 2^(n/2), where

c = 203.986136154799274492709451797084688042886818134781591... if n is even and

c = 201.491703180375661735217350021245093454724452720559762... if n is odd.

In closed form, a(n) ~ ((2 + sqrt(2)) * Product_{k>=3} (1/(1 - tau(k) / 2^(k/2))) + (-1)^n * (2 - sqrt(2)) * Product_{k>=3} (1/(1 - (-1)^k * tau(k) / 2^(k/2)))) * 2^(n/2 - 1), where tau() is A000005. (End)