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A032308 Expansion of Product_{k>=1} (1 + 3*x^k).

%I A032308 #36 Apr 12 2018 22:22:45
%S A032308 1,3,3,12,12,21,48,57,84,120,228,264,399,516,732,1119,1416,1884,2532,
%T A032308 3324,4296,6168,7545,9984,12684,16500,20577,26688,34572,43032,54264,
%U A032308 68232,84972,106176,131664,162507,205680,249888,308856,377796,465195,564024,691788,835572,1017768,1241040
%N A032308 Expansion of Product_{k>=1} (1 + 3*x^k).
%C A032308 "EFK" (unordered, size, unlabeled) transform of 3,3,3,3,...
%C A032308 Number of partitions into distinct parts of 3 sorts, see example. [_Joerg Arndt_, May 22 2013]
%H A032308 Vaclav Kotesovec, <a href="/A032308/b032308.txt">Table of n, a(n) for n = 0..10000</a>
%H A032308 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032308 G.f.: Product_{k>=1} (1 + 3*x^k).
%F A032308 a(n) = (1/4) * [x^n] QPochammer(-3, x). - _Vladimir Reshetnikov_, Nov 20 2015
%F A032308 a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/6 + log(3)^2/2 + polylog(2, -1/3) = 1.93937542076670895307727171917789144122... . - _Vaclav Kotesovec_, Jan 04 2016
%F A032308 G.f.: Sum_{i>=0} 3^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - _Ilya Gutkovskiy_, Apr 12 2018
%e A032308 From _Joerg Arndt_, May 22 2013: (Start)
%e A032308 There are a(5) = 21 partitions of 5 into distinct parts of 3 sorts (format P:S for part:sort):
%e A032308 01:  [ 1:0  4:0  ]
%e A032308 02:  [ 1:0  4:1  ]
%e A032308 03:  [ 1:0  4:2  ]
%e A032308 04:  [ 1:1  4:0  ]
%e A032308 05:  [ 1:1  4:1  ]
%e A032308 06:  [ 1:1  4:2  ]
%e A032308 07:  [ 1:2  4:0  ]
%e A032308 08:  [ 1:2  4:1  ]
%e A032308 09:  [ 1:2  4:2  ]
%e A032308 10:  [ 2:0  3:0  ]
%e A032308 11:  [ 2:0  3:1  ]
%e A032308 12:  [ 2:0  3:2  ]
%e A032308 13:  [ 2:1  3:0  ]
%e A032308 14:  [ 2:1  3:1  ]
%e A032308 15:  [ 2:1  3:2  ]
%e A032308 16:  [ 2:2  3:0  ]
%e A032308 17:  [ 2:2  3:1  ]
%e A032308 18:  [ 2:2  3:2  ]
%e A032308 19:  [ 5:0  ]
%e A032308 20:  [ 5:1  ]
%e A032308 21:  [ 5:2  ]
%e A032308 (End)
%p A032308 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A032308       `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, 3*b(n-i, i-1))))
%p A032308     end:
%p A032308 a:= n-> b(n$2):
%p A032308 seq(a(n), n=0..60);  # _Alois P. Heinz_, Aug 24 2015
%p A032308 # Alternatively:
%p A032308 simplify(expand(QDifferenceEquations:-QPochhammer(-3,x,99),x)/4):
%p A032308 seq(coeff(%,x,n), n=0..45); # _Peter Luschny_, Nov 17 2016
%t A032308 nmax = 40; CoefficientList[Series[Product[1 + 3*x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 24 2015 *)
%t A032308 nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*3^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 25 2015 *)
%t A032308 (QPochhammer[-3, x]/4 + O[x]^58)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)
%o A032308 (PARI) N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+3*x^n)) \\ _Joerg Arndt_, May 22 2013
%Y A032308 Cf. A000009, A032302, A261568, A261569.
%K A032308 nonn
%O A032308 0,2
%A A032308 _Christian G. Bower_
%E A032308 a(0) prepended and more terms added by _Joerg Arndt_, May 22 2013