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%I #16 May 03 2020 08:03:12
%S 1,1,1,4,9,14,19,24,39,63,87,111,155,235,329,423,552,771,1091,1430,
%T 1825,2400,3295,4392,5597,7117,9367,12476,16077,20182,25677,33472,
%U 43406,54578,68109,86475,111316,140965,174836,217520,275130,348555,433578,533640,662620,831747
%N a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k))^a(k).
%C a(n) = number of partitions of n into preceding terms starting from a(1), a(2), a(3), ... (for n > 1), with a(1) type of part a(1), a(2) types of part a(2), ...
%H Alois P. Heinz, <a href="/A293807/b293807.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 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))^a(n).
%e a(3) = 4 because we have [1a, 1a, 1a], [1a, 1a, 1b], [1a, 1b, 1b] and [1b, 1b, 1b].
%e G.f. = -x - 2*x^2 + 1/((1 - x)*(1 - x)*(1 - x^4)^4*(1 - x^9)^9*(1 - x^14)^14*(1 - x^19)^19*(1 - x^24)^24*(1 - x^39)^39*...) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 14*x^5 + 19*x^6 + 24*x^7 + 39*x^8 + ...
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-a(i)*j, i-1)*binomial(a(i)+j-1, j), j=0..n/a(i))))
%p end:
%p a:= n-> `if`(n<2, 1, b(n, n-1)):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Oct 16 2017
%t a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^a[k])^a[k], {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 45}]
%o (Python)
%o from sympy import binomial
%o from sympy.core.cache import cacheit
%o @cacheit
%o def b(n, i): return 1 if n==0 else 0 if i<1 else sum(b(n - a(i)*j, i - 1) * binomial(a(i) + j - 1, j) for j in range(n//a(i) + 1))
%o def a(n): return 1 if n<2 else b(n, n - 1)
%o print([a(n) for n in range(51)]) # _Indranil Ghosh_, Dec 13 2017, after Maple code
%Y Cf. A000081, A000219, A151945, A293806.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Oct 16 2017
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