|
|
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
|
|
|
FORMULA
|
G.f.: Product_{k>=1} (1 + 3*x^k).
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
|
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):
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-3, x, 99), x)/4):
|
|
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 *)
|
|
PROG
|
(PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+3*x^n)) \\ Joerg Arndt, May 22 2013
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
a(0) prepended and more terms added by Joerg Arndt, May 22 2013
|
|
STATUS
|
approved
|
|
|
|