A022568 Expansion of Product_{m>=1} (1+x^m)^3.
1, 3, 6, 13, 24, 42, 73, 120, 192, 302, 465, 702, 1046, 1536, 2226, 3195, 4536, 6378, 8896, 12306, 16896, 23045, 31224, 42048, 56310, 75000, 99384, 131072, 172071, 224910, 292774, 379608, 490338, 631104, 809472, 1034814, 1318707, 1675344, 2122176, 2680602, 3376728, 4242432, 5316562, 6646272
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
- Mohammed L. Nadji and Moussa Ahmia, Congruences for L-regular tripartitions for L in {2, 3}, Integers (2024) Vol. 24, Art. No. A86. See p. 2.
Programs
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Magma
Coefficients(&*[(1+x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
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Mathematica
nmax=50; CoefficientList[Series[Product[(1+q^m)^3,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *) nmax = 50; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 3; poly[[3]] = 3; poly[[4]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];, {p, 1, 3}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Mar 31 2018 *) -
PARI
x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^3)) \\ Indranil Ghosh, Apr 03 2017
Formula
a(n) ~ exp(Pi * sqrt(n)) / (8 * n^(3/4)) * (1 + (Pi/16 - 3/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017
a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
G.f.: Sum_{n >= 0} q^(n*(n+1)/2) / Sum_{n in Z} (-1)^n * q^(n^2). - Peter Bala, Jan 07 2025
Comments