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A101277
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Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.
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3
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1, 2, 3, 6, 10, 16, 25, 38, 57, 84, 121, 172, 243, 338, 465, 636, 862, 1158, 1546, 2050, 2702, 3542, 4616, 5986, 7729, 9932, 12707, 16196, 20563, 26010, 32788, 41194, 51591, 64418, 80195, 99558, 123269, 152226, 187514, 230434, 282519, 345596, 421844, 513834
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OFFSET
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0,2
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COMMENTS
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This is also A080054 times 1/Product_{k>=1} (1 - x^(2k)).
There are no partitions of 2n+1 in which all odd parts occur with multiplicity 2. - Michael Somos, Oct 27 2008
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LINKS
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FORMULA
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Euler transform of period 4 sequence [2, 0, 2, 1, ...]. - Michael Somos, Feb 10 2005
G.f.: (1/theta_4(0, x))*Product_{k>0}(1+x^(2k)) = theta_4(0, x^2)/theta_4(0, x)*Product_{k>0}(1-x^(2k)) = 1/Product_{k>0} ((1-x^(2k-1))^2 * (1-x^(4k))).
Expansion of 1 / (psi(-x) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Oct 27 2008
Expansion of q^(1/12) * eta(q^2)^2 / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Oct 27 2008
a(n) ~ sqrt(5) * exp(Pi*sqrt(5*n/6)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Aug 30 2015
G.f.: 2/((x; x)_inf * (-1; -x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 22 2016
Expansion of phi(-x^2) / f(-x)^2 = chi(x) / f(-x) = 1 / (chi(-x)^2 * f(-x^4)) = f(-x^4) / psi(-x)^2 = psi(-x) / chi(-x) = chi(x)^2 / psi(-x^2) in powers of x. - Michael Somos, Nov 22 2016
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EXAMPLE
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G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 57*x^8 + ...
G.f. = 1/q + 2*q^11 + 3*q^23 + 6*q^35 + 10*q^47 + 16*q^59 + 25*q^71 + ...
E.g. 12 = 10 + 2 = 10 + 1 + 1 = 8 + 4 = 8 + 2 + 2 = 8 + 2 + 1 + 1 = 6 + 6 = 6 + 4 + 2 = 6 + 4 + 1 + 1 = 6 + 3 + 3 = 6 + 2 + 2 + 2 = 6 + 2 + 2 + 1 + 1 = 5 + 5 + 2 = 5 + 5 + 1 + 1 = 4 + 4 + 4 = 4 + 4 + 2 + 2 = 4 + 4 + 2 + 1 + 1 = 4 + 3 + 3 + 2 = 4 + 3 + 3 + 1 + 1 = 4 + 2 + 2 + 2 + 2 = 4 + 2 + 2 + 2 + 1 + 1 = 3 + 3 + 2 + 2 + 2 = 3 + 3 + 2 + 2 + 1 + 1 = 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 + 1 + 1.
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MAPLE
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series(product(1/((1-x^(2*k-1))^2*(1-x^(4*k))), k=1..100), x=0, 100);
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MATHEMATICA
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nmax=50; CoefficientList[Series[Product[1/((1-x^(2*k-1))^2 * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, -x] QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 10 2005 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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