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A032308 Expansion of Product_{k>=1} (1 + 3*x^k). 26
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
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
G.f.: Sum_{i>=0} 3^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018
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
Sequence in context: A303309 A268774 A240801 * A117856 A074850 A073055
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(0) prepended and more terms added by Joerg Arndt, May 22 2013
STATUS
approved

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Last modified May 23 18:13 EDT 2024. Contains 372765 sequences. (Running on oeis4.)