OFFSET
0,4
COMMENTS
a(n) = number of partitions of n into preceding terms starting from a(1), a(2), a(3), ... (for n > 1).
LINKS
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)).
EXAMPLE
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 + ...
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 16 2017
STATUS
approved