A293806 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k)).

1, 1, 1, 4, 6, 8, 11, 14, 19, 24, 30, 37, 47, 57, 70, 84, 102, 121, 144, 170, 202, 235, 275, 319, 372, 429, 495, 567, 652, 742, 848, 963, 1095, 1237, 1399, 1574, 1775, 1990, 2235, 2499, 2795, 3114, 3473, 3859, 4292, 4755, 5271, 5827, 6444, 7107, 7840, 8625, 9493, 10422, 11444, 12541

Offset: 0

Author

Ilya Gutkovskiy, Oct 16 2017

Keywords

• nonn

Comments

a(n) = number of partitions of n into preceding terms starting from a(1), a(2), a(3), ... (for n > 1).

Examples

			a(4) = 6 because we have [4], [1a, 1a, 1a, 1a], [1a, 1a, 1a, 1b], [1a, 1a, 1b, 1b],  [1a, 1b, 1b, 1b] and [1b, 1b, 1b, 1b].
G.f. = -x - 2*x^2 + 1/((1 - x)*(1 - x)*(1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^11)*(1 - x^14)*(1 - x^19)*...) = 1 + x + x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 14*x^7 + 19*x^8 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A000041, A007279, A151945, A229362.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(a(i)>n, 0, b(n-a(i), i))))
    end:
a:= n-> `if`(n<2, 1, b(n, n-1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2017

Mathematica

a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 55}]

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} 1/(1 - x^a(n)).