OFFSET
0,3
COMMENTS
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
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
For the name "cubic partition" see Xiong; Chen & Lin; Chern & Dastidar. - Michel Marcus, Jan 28 2016
Also the number of nonzero terms in the power sum polysymmetric function expansion of the unsigned elementary polysymmetric function E^+_n. - Aditya Khanna, Nov 26 2025
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0930 and N0931).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Zakir Ahmed, Nayandeep Deka Baruah, and Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Koustav Banerjee, New Asymptotics and Inequalities Related to the Partition Function, Doctoral Thesis, Johannes Kepler Univ. (Linz, Austria 2022).
Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, and WenHuan Zeng, New inequalities for p(n) and log p(n), Ramanujan J. 61 (2023) 1295-1338. See pp. 1297.
M. F. Capobianco and C. F. Pinzka, Problem 2055, Amer. Math. Monthly, 75 (1968), 188; 76 (1969), 194.
William Y. C. Chen and Bernard L. S. Lin, Congruences for the Number of Cubic Partitions Derived from Modular Forms, arXiv:0910.1263 [math.NT], 2016.
Shane Chern and Manosij Ghosh Dastidar, Congruences and recursions for the cubic partitions, arXiv:1601.06480 [math.NT], 2016.
Marston Conder, Tomaš Pisanski, and Arjana Žitnik, Vertex-transitive graphs and their arc-types, arXiv preprint arXiv:1505.02029 [math.CO], 2015.
R. K. Guy, Letter to Morris Newman, Aug 21 1986, concerning A2513 (annotated scanned copy, with permission).
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.
Aditya Khanna, Transition Matrices between Plethystic Bases of Polysymmetric Functions via Bijective Methods, arXiv:2510.12723 [math.CO], 2025. See pp. 5, 46.
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. 16.
Vaclav Kotesovec, Asymptotics of sequence A002513, 2019.
Lukas Mauth, Exact formula for cubic partitions, arXiv:2305.03396 [math.NT], 2023.
Morris Newman, Construction and application of a class of modular functions (II). Proc. London Math. Soc. (3) 9 1959 373-387.
Morris Newman, Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
James A. Sellers, Elementary proofs of congruences for the cubic and overcubic partition functions, Australasian Journal of Combinatorics, Volume 60(2) (2014), Pages 191-197.
Xinhua Xiong, The number of cubic partitions modulo powers of 5, arXiv:1004.4737 [math.NT], 2010.
FORMULA
From Michael Somos, Mar 23 2003: (Start)
Expansion of q^(1/8) / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 2 sequence [1, 2, ...].
G.f.: Product_{k>0} 1/((1 - x^(2*k))^2 * (1 - x^(2*k-1))).
(End)
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
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
From Michel Marcus, Jan 28 2016: (Start)
G.f.: Product_{k>0} 1/((1 - x^k) * (1 - x^(2*k))).
a(3n+2) = 0 (mod 3).
a(25n+22) = 0 (mod 5) (see Xiong).
a(49n+15) = a(49n+29) = a(49n+36) = a(49n+43) = 0 (mod 7) (see Chen & Lin).
a(297n+62) = a(297n+161) = 0 (mod 11) (see Chern & Dastidar).
(End)
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
G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018
From Peter Bala, Sep 25 2023: (Start)
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.
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.
The self-convolution A(x)^2 is the g.f. of A319455. (End)
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (exp(-Pi/8) / sqrt(Pi)) * 2^(7/8) * Gamma(3/4)^2 = A388109. - Simon Plouffe, Sep 15 2025
EXAMPLE
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 + ...
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 + ...
MAPLE
N:= 50: # to get a(0) to a(N)
P:= mul((1-x^(2*k))^(-2)*(1-x^(2*k-1))^(-1), k=1..ceil(N/2)):
S:= series(P, x, N+1):
seq(coeff(S, x, j), j=0..N); # Robert Israel, Jan 26 2016
# Alternative:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
`if`(d::odd, d, 2*d), d=numtheory[divisors](j)), j=1..n)/n)
end:
seq(a(n), n=0..50); # Alois P. Heinz, Nov 05 2020
MATHEMATICA
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 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q] / QPochhammer[ q^2], {q, 0, n}]; (* Michael Somos, Jul 17 2013 *)
Table[Sum[PartitionsP[k]*PartitionsP[n-2k], {k, 0, n/2}], {n, 0, 50}] (* Vaclav Kotesovec, Jun 22 2015 *)
PROG
(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 */
(SageMath) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 2)
a = EulerTransform(b)
print([a(n) for n in range(44)]) # Peter Luschny, Nov 17 2022
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms and information from Michael Somos, Mar 23 2003
STATUS
approved
