OFFSET
0,2
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
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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
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
EXAMPLE
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)
MAPLE
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, b(n, i-1)+`if`(i>n, 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 *)
PROG
(PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+2*x^n)) \\ Joerg Arndt, May 22 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian G. Bower, Apr 01 1998
STATUS
approved