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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 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 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)/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 *) 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: A303309 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 May 23 18:13 EDT 2024. Contains 372765 sequences. (Running on oeis4.)