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

1, 2, 2, 6, 6, 10, 18, 22, 30, 42, 66, 78, 110, 138, 186, 254, 318, 402, 522, 654, 822, 1074, 1306, 1638, 2022, 2514, 3058, 3798, 4662, 5658, 6882, 8358, 10062, 12186, 14610, 17534, 21150, 25146, 29994, 35694, 42446, 50178, 59514, 70110, 82758, 97602, 114570, 134262

Offset: 0

Author

Christian G. Bower, Apr 01 1998

Keywords

• nonn

Comments

"EFK" (unordered, size, unlabeled) transform of 2,2,2,2,...

Number of partitions into distinct parts of 2 sorts, see example. - Joerg Arndt, May 22 2013

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

Antidiagonal sums of A284593. - Peter Bala, Mar 30 2017

Examples

			From _Joerg Arndt_, May 22 2013: (Start)
There are a(7) = 22 partitions of 7 into distinct parts of 2 sorts (format P:S for part:sort):
01:  [ 1:0  2:0  4:0  ]
02:  [ 1:0  2:0  4:1  ]
03:  [ 1:0  2:1  4:0  ]
04:  [ 1:0  2:1  4:1  ]
05:  [ 1:0  6:0  ]
06:  [ 1:0  6:1  ]
07:  [ 1:1  2:0  4:0  ]
08:  [ 1:1  2:0  4:1  ]
09:  [ 1:1  2:1  4:0  ]
10:  [ 1:1  2:1  4:1  ]
11:  [ 1:1  6:0  ]
12:  [ 1:1  6:1  ]
13:  [ 2:0  5:0  ]
14:  [ 2:0  5:1  ]
15:  [ 2:1  5:0  ]
16:  [ 2:1  5:1  ]
17:  [ 3:0  4:0  ]
18:  [ 3:0  4:1  ]
19:  [ 3:1  4:0  ]
20:  [ 3:1  4:1  ]
21:  [ 7:0  ]
22:  [ 7:1  ]
(End)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Moussa Ahmia and Mohammed L. Nadji, Overpartitions into distinct parts, Comb. Number Theory 14 (2025), no. 1, 65-74, preprint (2021).
• C. G. Bower, Transforms (2)
• Vaclav Kotesovec, Asymptotic formula for A032302
• Eric Weisstein's World of Mathematics, Dilogarithm
• Eric Weisstein's MathWorld, Polylogarithm
• Wikipedia, Polylogarithm

Crossrefs

Cf. A000009, A032308, A261562, A261568, A261569, A266576, A284593.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-2,x,99)/3,x)):
seq(coeff(%,x,n), n=0..47); # Peter Luschny, Nov 17 2016

Mathematica

nn=47; CoefficientList[Series[Product[1+2x^i,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 07 2013 *)
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 *)
(QPochhammer[-2, x]/3 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

PARI

N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+2*x^n)) \\ Joerg Arndt, May 22 2013

Formula

a(n) = A072706(n)*2 for n>=1.

G.f.: Sum_{n>=0} (2^n*q^(n*(n+1)/2) / Product_{k=1..n} (1-q^k ) ). - Joerg Arndt, Jan 20 2014

a(n) = (1/3) [x^n] QPochhammer(-2,x). - Vladimir Reshetnikov, Nov 20 2015

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