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

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

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

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]

Examples

			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)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A000009, A032302, A261568, A261569.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 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 *)

PARI

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

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

G.f.: Sum_{i>=0} 3^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

Extensions

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