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A098151
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Number of partitions of 2n prime to 3 with all odd parts occurring with even multiplicities. There is no restriction on the even parts.
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20
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1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 198, 268, 360, 480, 634, 832, 1084, 1404, 1808, 2316, 2952, 3744, 4728, 5946, 7448, 9294, 11556, 14320, 17688, 21780, 26740, 32736, 39968, 48672, 59122, 71644, 86616, 104484, 125768, 151072, 181104, 216684
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OFFSET
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0,2
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COMMENTS
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There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - Michael Somos, Apr 15 2012
a(n) is also the number of Schur overpartitions of n, i.e., the number of overpartitions of n where adjacent parts differ by at least 3 if the smaller is overlined or divisible by 3 and adjacent parts differ by at least 6 if the smaller is overlined and divisible by 3. - Jeremy Lovejoy, Mar 23 2015
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LINKS
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FORMULA
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Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 15 2012
Expansion of f(q, q^2) / f(-q, -q^2) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 15 2012
Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
G.f. = (Sum((-1)^n*q^(3*n^2),n=-oo..oo)) /(Sum((-1)^n*q^(n^2),n=-oo..oo)). - N. J. A. Sloane, Aug 09 2016
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u^2) * (u^2 + v^4) - 4 * u^2*v^4. - Michael Somos, Apr 15 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3 * u*v^2 - 3 * u^2*v^3. - Michael Somos, Dec 04 2004
Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - Vladeta Jovovic, Sep 24 2004
Taylor series of product_{k=1..inf}(1+x^k+x^(2*k))/(1-x^k+x^(2*k))= product_{k=1..inf}(1+x^k)(1-x^(3k))/((1-x^k)(1+x^(3k)))=Theta_4(0, x^3)/theta_4(0, x)
a(n) ~ Pi * BesselI(1, Pi*sqrt(2*n/3)) / (3*sqrt(2*n)) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)) * (1 - 3*sqrt(3)/(8*Pi*sqrt(2*n)) - 45/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 09 2017
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EXAMPLE
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E.g a(4)=10 because 8=4+4=4+2+2=2+2+2+2=2+2+2+1+1=2+2+1+1+1+1=4+2+1+1=4+1+1+1+1=2+1+1+1+1=1+1+1+1+1+1+1+1=...
G.f. = 1 + 2*q + 4*q^2 + 6*q^3 + 10*q^4 + 16*q^5 + 24*q^6 + 36*q^7 + 52*q^8 + ...
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MAPLE
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series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)), k=1..150), x=0, 100);
# alternative program using expansion of f(q, q^2) / f(-q, -q^2):
with(gfun): series( add(x^(n*(3*n-1)/2), n = -8..8)/add((-1)^n*x^(n*(3*n-1)/2), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* Michael Somos, Oct 23 2013 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1)) * (1+x^(3*k-2)) / ((1-x^(3*k-1)) * (1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Dec 04 2004 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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