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A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

%I A281425 #37 Mar 08 2025 06:03:17
%S A281425 1,0,1,-1,2,-4,9,-21,49,-112,249,-539,1143,-2396,5013,-10550,22420,
%T A281425 -48086,103703,-223806,481388,-1029507,2187944,-4625058,9742223,
%U A281425 -20490753,43111808,-90840465,191773014,-405523635,858378825,-1817304609,3845492204,-8129023694,17162802918,-36191083386
%N A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).
%C A281425 a(n) is n-th term of the Euler transform of -n + 1, 1, 1, 1, ...
%C A281425 Inverse zero-based binomial transform of A000041. The version for strict partitions is A380412, or A293467 up to sign. - _Gus Wiseman_, Feb 06 2025
%H A281425 Vaclav Kotesovec, <a href="/A281425/b281425.txt">Table of n, a(n) for n = 0..3000</a>
%H A281425 A. M. Odlyzko, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4932.pdf">Differences of the partition function</a>, Acta Arithmetica 49.3 (1988): 237-254.
%H A281425 Dennis Stanton and Doron Zeilberger, <a href="https://doi.org/10.1090/S0002-9939-1989-0972238-1">The Odlyzko conjecture and O’Hara’s unimodality proof</a>, Proceedings of the American Mathematical Society 107.1 (1989): 39-42.
%F A281425 a(n) = [q^n] 1/((1 + q)*(1 + q + q^2)*...*(1 + q + ... + q^(n-1))).
%F A281425 a(n) = Sum_{j=0..n} (-1)^j * binomial(n, j) * A000041(n-j). - _Vaclav Kotesovec_, Oct 06 2017
%F A281425 a(n) ~ (-1)^n * 2^(n - 3/2) * exp(Pi*sqrt(n/12) + Pi^2/96) / (sqrt(3)*n). - _Vaclav Kotesovec_, May 07 2018
%p A281425 b:= proc(n, k) option remember; `if`(k=0,
%p A281425       combinat[numbpart](n), b(n, k-1)-b(n-1, k-1))
%p A281425     end:
%p A281425 a:= n-> b(n$2):
%p A281425 seq(a(n), n=0..35);  # _Alois P. Heinz_, Dec 21 2024
%t A281425 Table[SeriesCoefficient[(1 - q)^n / Product[(1 - q^j), {j, 1, n}], {q, 0, n}], {n, 0, 35}]
%t A281425 Table[SeriesCoefficient[(1 - q)^n QPochhammer[q^(1 + n), q]/QPochhammer[q, q], {q, 0, n}], {n, 0, 35}]
%t A281425 Table[SeriesCoefficient[1/QFactorial[n, q], {q, 0, n}], {n, 0, 35}]
%t A281425 Table[Differences[PartitionsP[Range[0, n]], n], {n, 0, 35}] // Flatten
%t A281425 Table[Sum[(-1)^j*Binomial[n, j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Oct 06 2017 *)
%Y A281425 Cf. A128566, A292463, A292541, A292613.
%Y A281425 Cf. A000041, A002865, A053445, A275638, A275639, A275640, A275641, A275642, A275643, A275644.
%Y A281425 Cf. A218481, A294466, A095051.
%Y A281425 Cf. A266232, A294467, A293467, A294468.
%Y A281425 Row n=0 of A175804 (row n=1 is A320590 except first term).
%Y A281425 Cf. A000009, A008284, A087897, A116608, A129519, A225486, A320591, A377056, A378622.
%K A281425 sign
%O A281425 0,5
%A A281425 _Ilya Gutkovskiy_, Oct 05 2017