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A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

%I A032302 #70 Aug 16 2025 15:30:16
%S A032302 1,2,2,6,6,10,18,22,30,42,66,78,110,138,186,254,318,402,522,654,822,
%T A032302 1074,1306,1638,2022,2514,3058,3798,4662,5658,6882,8358,10062,12186,
%U A032302 14610,17534,21150,25146,29994,35694,42446,50178,59514,70110,82758,97602,114570,134262
%N A032302 G.f.: Product_{k>=1} (1 + 2*x^k).
%C A032302 "EFK" (unordered, size, unlabeled) transform of 2,2,2,2,...
%C A032302 Number of partitions into distinct parts of 2 sorts, see example. - _Joerg Arndt_, May 22 2013
%C A032302 In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - _Vaclav Kotesovec_, Jan 04 2016
%C A032302 Antidiagonal sums of A284593. - _Peter Bala_, Mar 30 2017
%H A032302 Vaclav Kotesovec, <a href="/A032302/b032302.txt">Table of n, a(n) for n = 0..10000</a>
%H A032302 Moussa Ahmia and Mohammed L. Nadji, <a href="https://doi.org/10.2140/cnt.2025.14.65">Overpartitions into distinct parts</a>, Comb. Number Theory 14 (2025), no. 1, 65-74, <a href="https://doi.org/10.21203/rs.3.rs-2223458/v1">preprint (2021)</a>.
%H A032302 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032302 Vaclav Kotesovec, <a href="/A032302/a032302.pdf">Asymptotic formula for A032302</a>
%H A032302 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>
%H A032302 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>
%H A032302 Wikipedia, <a href="http://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>
%F A032302 a(n) = A072706(n)*2 for n>=1.
%F A032302 G.f.: Sum_{n>=0} (2^n*q^(n*(n+1)/2) / Product_{k=1..n} (1-q^k ) ). - _Joerg Arndt_, Jan 20 2014
%F A032302 a(n) = (1/3) [x^n] QPochhammer(-2,x). - _Vladimir Reshetnikov_, Nov 20 2015
%F A032302 a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/6 + log(2)^2/2 + polylog(2, -1/2) = 1.43674636688368094636290202389358335424... . Equivalently, c = A266576 = Pi^2/12 + log(2)^2 + polylog(2, 1/4)/2. - _Vaclav Kotesovec_, Jan 04 2016
%e A032302 From _Joerg Arndt_, May 22 2013: (Start)
%e A032302 There are a(7) = 22 partitions of 7 into distinct parts of 2 sorts (format P:S for part:sort):
%e A032302 01:  [ 1:0  2:0  4:0  ]
%e A032302 02:  [ 1:0  2:0  4:1  ]
%e A032302 03:  [ 1:0  2:1  4:0  ]
%e A032302 04:  [ 1:0  2:1  4:1  ]
%e A032302 05:  [ 1:0  6:0  ]
%e A032302 06:  [ 1:0  6:1  ]
%e A032302 07:  [ 1:1  2:0  4:0  ]
%e A032302 08:  [ 1:1  2:0  4:1  ]
%e A032302 09:  [ 1:1  2:1  4:0  ]
%e A032302 10:  [ 1:1  2:1  4:1  ]
%e A032302 11:  [ 1:1  6:0  ]
%e A032302 12:  [ 1:1  6:1  ]
%e A032302 13:  [ 2:0  5:0  ]
%e A032302 14:  [ 2:0  5:1  ]
%e A032302 15:  [ 2:1  5:0  ]
%e A032302 16:  [ 2:1  5:1  ]
%e A032302 17:  [ 3:0  4:0  ]
%e A032302 18:  [ 3:0  4:1  ]
%e A032302 19:  [ 3:1  4:0  ]
%e A032302 20:  [ 3:1  4:1  ]
%e A032302 21:  [ 7:0  ]
%e A032302 22:  [ 7:1  ]
%e A032302 (End)
%p A032302 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A032302       `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, 2*b(n-i, i-1))))
%p A032302     end:
%p A032302 a:= n-> b(n$2):
%p A032302 seq(a(n), n=0..60);  # _Alois P. Heinz_, Aug 24 2015
%p A032302 # Alternatively:
%p A032302 simplify(expand(QDifferenceEquations:-QPochhammer(-2,x,99)/3,x)):
%p A032302 seq(coeff(%,x,n), n=0..47); # _Peter Luschny_, Nov 17 2016
%t A032302 nn=47; CoefficientList[Series[Product[1+2x^i,{i,1,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 07 2013 *)
%t A032302 nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 25 2015 *)
%t A032302 (QPochhammer[-2, x]/3 + O[x]^58)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)
%o A032302 (PARI) N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+2*x^n)) \\ _Joerg Arndt_, May 22 2013
%Y A032302 Cf. A000009, A032308, A261562, A261568, A261569, A266576, A284593.
%K A032302 nonn
%O A032302 0,2
%A A032302 _Christian G. Bower_, Apr 01 1998