A273225 -id:A273225 - cp's OEIS frontend

A002513 Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.

1, 1, 3, 4, 9, 12, 23, 31, 54, 73, 118, 159, 246, 329, 489, 651, 940, 1242, 1751, 2298, 3177, 4142, 5630, 7293, 9776, 12584, 16659, 21320, 27922, 35532, 46092, 58342, 75039, 94503, 120615, 151173, 191611, 239060, 301086, 374026, 468342, 579408, 721638, 889287

Offset: 0

N. J. A. Sloane, Simon Plouffe

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

			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 + ...

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).

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).
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.
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.

Cf. A000122, A015128, A001934, A089799, A215947, A273225, A319455.

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
# second Maple program:
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

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 *)

{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 */

# 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

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)

More terms and information from Michael Somos, Mar 23 2003

A274621 Coefficients in the expansion Product_{ n>=1 } (1-q^(2n-1))^2/(1-q^(2n))^2.

Original entry on oeis.org

1, -2, 3, -6, 11, -18, 28, -44, 69, -104, 152, -222, 323, -460, 645, -902, 1254, -1722, 2343, -3174, 4278, -5722, 7601, -10056, 13250, -17358, 22623, -29382, 38021, -48984, 62857, -80404, 102528, -130282, 165002, -208398, 262495, -329666, 412878, -515840, 642941, -799362, 991478

Offset: 0

N. J. A. Sloane, Jul 03 2016

This is the reciprocal of the g.f. for A008441.
From Wolfdieter Lang, Jul 05 2016: (Start)
The g.f. is the square of the one for A106507.
Expansion of 1/(k/(4*q^(1/2)) * (2/Pi)*K(k)) in powers of q^2, where k is the modulus (k^2 is the parameter), K is the real quarter period and q is the Jacobi nome of elliptic functions. See a similar Jul 05 2016 comment on A008441. This appears as a factor in the sn and cn formulas of Abramowitz-Stegun. p. 575, 16.23.1 and 16.23.2. (End)

R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts

If the signs are deleted we get A273225.

Cf. A008441, A010054, A106507.

nmax = 40; CoefficientList[Series[Product[(1 - x^(2*k-1))^2 / (1 - x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2016 *)

From Wolfdieter Lang, Jul 05 2016: (Start)
G.f.: 1/(theta_2(0, sqrt(q))/(2*q^(1/8)))^2, with the Jacobi theta_2 function.
G.f.: 1/(Sum_{n >= 0} q^(n*(n+1)/2))^2.
G.f.: 1/(Prod_{n >= 1} (1 - q^n) * (1 + q^n)^2)^2 = 1/(Prod_{n >= 1} (1 - q^(2*n)) * (1 + q^n ))^2 = Prod_{n >= 1} (1 - q^(2n-1))^2 / (1 - q^(2n))^2. For the last equality, giving the g.f. of the name, see the Euler identity, mentioned in a Jul 05 2016 comment of A010054. (End)
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 55, 231, 981, 4222, 18351, 80320, 353453, 1562364, 6932185, 30856541, 137725710, 616190583, 2762605791, 12408541299, 55825435656, 251523510045, 1134741006825, 5125453110196, 23175983361270, 104899547541255, 475228898015025, 2154737528486881, 9777332125043577

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A006950, A106337, A192540, A273225, A273226, A273228, A295832, A296043, A296044.

Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(2 (-x)^(1/8)/EllipticTheta[2, 0, Sqrt[-x]])^n, {x, 0, n}], {n, 0, 26}]
Table[(-1)^n * 2^n * SeriesCoefficient[1/(QPochhammer[-1, x]*QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 07 2020 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 4/Sqrt[Pi*(77/2 - 4*s*(-r*s)^(7/8) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-r*s]])]} /. FindRoot[{s == (2*(-r*s)^(1/8))/EllipticTheta[2, 0, Sqrt[-r*s]], 7*I*r + 2*(-r*s)^(7/8)*Sqrt[r*s] * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-r*s]] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^(4*k)))^n.

a(n) ~ c * d^n / sqrt(n), where d = 4.62579056836776492108784045382518984897... (see A192540) and c = 0.255113338880004277664416308115912337... - Vaclav Kotesovec, Dec 05 2017

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 20, 33, 50, 74, 114, 175, 257, 375, 555, 814, 1171, 1677, 2406, 3435, 4855, 6825, 9591, 13428, 18667, 25851, 35745, 49250, 67544, 92340, 125966, 171345, 232257, 313945, 423470, 569778, 764465, 1023231, 1366827, 1821756, 2422394, 3214318, 4257088, 5627086, 7422941

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A000219, A006950, A113415, A156616, A224364, A263140, A273225, A273226, A274621, A295831.

nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[x^k ((-1)^(k + 1) + x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} x^k*((-1)^(k+1) + x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3 * (7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (24 * (7*Zeta(3))^(1/3)) - Pi^4 / (12096 * Zeta(3)) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(23/24) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

cp's OEIS Frontend

A002513 Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A274621 Coefficients in the expansion Product_{ n>=1 } (1-q^(2n-1))^2/(1-q^(2n))^2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A295832 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^n.

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.