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A032308 Expansion of Product_{k>=1} (1 + 3*x^k). 13
1, 3, 3, 12, 12, 21, 48, 57, 84, 120, 228, 264, 399, 516, 732, 1119, 1416, 1884, 2532, 3324, 4296, 6168, 7545, 9984, 12684, 16500, 20577, 26688, 34572, 43032, 54264, 68232, 84972, 106176, 131664, 162507, 205680, 249888, 308856, 377796, 465195, 564024, 691788, 835572, 1017768, 1241040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

"EFK" (unordered, size, unlabeled) transform of 3,3,3,3,...

Number of partitions into distinct parts of 3 sorts, see example. [Joerg Arndt, May 22 2013]

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

C. G. Bower, Transforms (2)

FORMULA

G.f.: Product_{k>=1} (1 + 3*x^k).

a(n) = (1/4) * [x^n] QPochammer(-3, x). - Vladimir Reshetnikov, Nov 20 2015

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

EXAMPLE

From Joerg Arndt, May 22 2013: (Start)

There are a(5) = 21 partitions of 5 into distinct parts of 3 sorts (format P:S for part:sort):

01:  [ 1:0  4:0  ]

02:  [ 1:0  4:1  ]

03:  [ 1:0  4:2  ]

04:  [ 1:1  4:0  ]

05:  [ 1:1  4:1  ]

06:  [ 1:1  4:2  ]

07:  [ 1:2  4:0  ]

08:  [ 1:2  4:1  ]

09:  [ 1:2  4:2  ]

10:  [ 2:0  3:0  ]

11:  [ 2:0  3:1  ]

12:  [ 2:0  3:2  ]

13:  [ 2:1  3:0  ]

14:  [ 2:1  3:1  ]

15:  [ 2:1  3:2  ]

16:  [ 2:2  3:0  ]

17:  [ 2:2  3:1  ]

18:  [ 2:2  3:2  ]

19:  [ 5:0  ]

20:  [ 5:1  ]

21:  [ 5:2  ]

(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, 3*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(-3, x, 99), x)/4):

seq(coeff(%, x, n), n=0..45); # Peter Luschny, Nov 17 2016

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[1 + 3*x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2015 *)

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 *)

(QPochhammer[-3, x]/4 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

PROG

(PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+3*x^n)) \\ Joerg Arndt, May 22 2013

CROSSREFS

Cf. A000009, A032302, A261568, A261569.

Sequence in context: A065957 A268774 A240801 * A117856 A074850 A073055

Adjacent sequences:  A032305 A032306 A032307 * A032309 A032310 A032311

KEYWORD

nonn

AUTHOR

Christian G. Bower

EXTENSIONS

a(0) prepended and more terms added by Joerg Arndt, May 22 2013

STATUS

approved

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Last modified February 18 15:31 EST 2018. Contains 299324 sequences. (Running on oeis4.)