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A002513
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Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.
(Formerly M2354 N0930 N0931)
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21
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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M. F. Capobianco and C. F. Pinzka, Problem 2055, Amer. Math. Monthly, 75 (1968), 188; 76 (1969), 194.
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FORMULA
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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
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
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)
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EXAMPLE
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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 + ...
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MAPLE
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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):
# 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:
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MATHEMATICA
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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 *)
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PROG
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(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 */
(Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 2)
a = EulerTransform(b)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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