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%I M2354 N0930 N0931 #125 Feb 07 2024 09:01:35
%S 1,1,3,4,9,12,23,31,54,73,118,159,246,329,489,651,940,1242,1751,2298,
%T 3177,4142,5630,7293,9776,12584,16659,21320,27922,35532,46092,58342,
%U 75039,94503,120615,151173,191611,239060,301086,374026,468342,579408,721638,889287
%N Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.
%C For a real polynomial equation of degree n, a(n) is the number of possibilities for the roots to be real and unequal, real and equal (in various combinations), or simple or multiple complex conjugates. For example, a(3)=4 because we can have: three equal roots, two equal roots, three distinct real roots and two complex roots (see the Monthly Problem reference). - _Emeric Deutsch_, Mar 22 2005
%C Number of partitions of n, the even parts being of two kinds. E.g. a(4)=9 because we have 4, 4', 3+1, 2+2, 2+2', 2'+2', 2+1+1, 2'+1+1, 1+1+1+1. - _Emeric Deutsch_, Mar 22 2005
%C For the name "cubic partition" see Xiong; Chen & Lin; Chern & Dastidar. - _Michel Marcus_, Jan 28 2016
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0930 and N0931).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002513/b002513.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H Zakir Ahmed, Nayandeep Deka Baruah, and Manosij Ghosh Dastidar, <a href="https://doi.org/10.1016/j.jnt.2015.05.002">New congruences modulo 5 for the number of 2-color partitions</a>, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
%H Koustav Banerjee, <a href="https://epub.jku.at/obvulihs/content/titleinfo/8258752/">New Asymptotics and Inequalities Related to the Partition Function</a>, Doctoral Thesis, Johannes Kepler Univ. (Linz, Austria 2022).
%H M. F. Capobianco and C. F. Pinzka, <a href="http://www.jstor.org/stable/2317284">Problem 2055</a>, Amer. Math. Monthly, 75 (1968), 188; 76 (1969), 194.
%H William Y. C. Chen and Bernard L. S. Lin, <a href="http://arxiv.org/abs/0910.1263">Congruences for the Number of Cubic Partitions Derived from Modular Forms</a>, arXiv:0910.1263 [math.NT], 2016.
%H Shane Chern and Manosij Ghosh Dastidar, <a href="http://arxiv.org/abs/1601.06480">Congruences and recursions for the cubic partitions</a>, arXiv:1601.06480 [math.NT], 2016.
%H Marston Conder, Tomaš Pisanski, and Arjana Žitnik, <a href="http://arxiv.org/abs/1505.02029">Vertex-transitive graphs and their arc-types</a>, arXiv preprint arXiv:1505.02029 [math.CO], 2015.
%H R. K. Guy, <a href="/A002513/a002513.pdf">Letter to Morris Newman, Aug 21 1986</a>, concerning A2513 (annotated scanned copy, with permission).
%H David J. Hemmer, <a href="https://arxiv.org/abs/2402.03643">Generating functions for fixed points of the Mullineux map</a>, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 16.
%H Vaclav Kotesovec, <a href="/A002513/a002513_1.pdf">Asymptotics of sequence A002513</a>, 2019.
%H Lukas Mauth, <a href="https://arxiv.org/abs/2305.03396">Exact formula for cubic partitions</a>, arXiv:2305.03396 [math.NT], 2023.
%H Morris Newman, <a href="https://doi.org/10.1112/plms/s3-9.3.373">Construction and application of a class of modular functions (II)</a>. Proc. London Math. Soc. (3) 9 1959 373-387.
%H Morris Newman, <a href="/A002507/a002507.pdf">Construction and application of a class of modular functions, II</a>, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
%H James A. Sellers, <a href="https://ajc.maths.uq.edu.au/pdf/60/ajc_v60_p191.pdf">Elementary proofs of congruences for the cubic and overcubic partition functions</a>, Australasian Journal of Combinatorics, Volume 60(2) (2014), Pages 191-197.
%H Xinhua Xiong, <a href="http://arxiv.org/abs/1004.4737">The number of cubic partitions modulo powers of 5</a>, arXiv:1004.4737 [math.NT], 2010.
%F From _Michael Somos_, Mar 23 2003: (Start)
%F Expansion of q^(1/8) / (eta(q) * eta(q^2)) in powers of q.
%F Euler transform of period 2 sequence [1, 2, ...].
%F G.f.: Product_{k>0} 1/((1 - x^(2*k))^2 * (1 - x^(2*k-1))).
%F (End)
%F Given g.f. A(x), then B(q) = A(q)^8 / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 16*v^4 + v^3*w + 256*u*v^3 + 16*u*v^2*w - u^2*w^2. - _Michael Somos_, Apr 03 2005
%F a(n) ~ exp(Pi*sqrt(n)) / (8*n^(5/4)) * (1 - (Pi/16 + 15/(8*Pi))/sqrt(n)). - _Vaclav Kotesovec_, Jun 22 2015, extended Jan 17 2017
%F From _Michel Marcus_, Jan 28 2016: (Start)
%F G.f.: Product_{k>0} 1/((1 - x^k) * (1 - x^(2*k))).
%F a(3n+2) = 0 (mod 3).
%F a(25n+22) = 0 (mod 5) (see Xiong).
%F a(49n+15) = a(49n+29) = a(49n+36) = a(49n+43) = 0 (mod 7) (see Chen & Lin).
%F a(297n+62) = a(297n+161) = 0 (mod 11) (see Chern & Dastidar).
%F (End)
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(-7/2) (t/i)^-1 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Oct 17 2017
%F G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^(2*k)))). - _Ilya Gutkovskiy_, Aug 13 2018
%F From _Peter Bala_, Sep 25 2023: (Start)
%F The g.f. A(x) satisfies log(A(x)) = x + 5*x^2/2 + 4*x^3/3 + 13*x^4/4 + ... = Sum_{n >= 1} A215947(n)*x^n/n.
%F A(x^2) = 4/(F(x)*F(-x)) = 2/(F(x)*G(-x)), where F(x) = Sum_{n = -oo..oo} x^(n*(n+1)/2) is the g.f. of A089799 and G(x) = Sum_{n = -oo..oo} x^(n^2) is the g.f. of A000122. Cf. A001934. Note that 4/(F(-x)*F(-x)) is the g.f. of A273225.
%F The self-convolution A(x)^2 is the g.f. of A319455. (End)
%e G.f. = 1 + x + 3*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 23*x^6 + 31*x^7 + 54*x^8 + 73*x^9 + ...
%e G.f. = 1/q + q^7 + 3*q^15 + 4*q^23 + 9*q^31 + 12*q^39 + 23*q^47 + 31*q^55 + 54*q^63 + ...
%p N:= 50: # to get a(0) to a(N)
%p P:= mul((1-x^(2*k))^(-2)*(1-x^(2*k-1))^(-1),k=1..ceil(N/2)):
%p S:= series(P, x, N+1):
%p seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Jan 26 2016
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p `if`(d::odd, d, 2*d), d=numtheory[divisors](j)), j=1..n)/n)
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Nov 05 2020
%t max = 50; f[x_] := Product[ 1/((1-x^(2 k))^2*(1-x^(2k-1))), {k, 1, Ceiling[max/2]} ]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* _Jean-François Alcover_, Nov 04 2011 *)
%t a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q] / QPochhammer[ q^2], {q, 0, n}];(* _Michael Somos_, Jul 17 2013 *)
%t Table[Sum[PartitionsP[k]*PartitionsP[n-2k],{k,0,n/2}],{n,0,50}] (* _Vaclav Kotesovec_, Jun 22 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / eta(x + A) / eta(x^2 + A), n))}; /* _Michael Somos_, Nov 10 2005 */
%o (Sage) # uses[EulerTransform from A166861]
%o b = BinaryRecurrenceSequence(0, 1, 2)
%o a = EulerTransform(b)
%o print([a(n) for n in range(44)]) # _Peter Luschny_, Nov 17 2022
%Y Cf. A000122, A015128, A001934, A089799, A215947, A273225, A319455.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms and information from _Michael Somos_, Mar 23 2003
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