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A032302 G.f.: Product_{k>=1} (1 + 2*x^k). 25
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 (list; graph; refs; listen; history; text; internal format)
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)

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

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

Sequence in context: A237363 A082542 A162776 * A032214 A290261 A007040

Adjacent sequences:  A032299 A032300 A032301 * A032303 A032304 A032305

KEYWORD

nonn

AUTHOR

Christian G. Bower, Apr 01 1998

STATUS

approved

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Last modified October 17 19:18 EDT 2018. Contains 316293 sequences. (Running on oeis4.)