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A273225
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Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).
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5
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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
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OFFSET
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0,2
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COMMENTS
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Number of bipartitions of 'n' wherein odd parts are distinct (and even parts are unrestricted).
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LINKS
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FORMULA
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G.f.: Product_{k>=1} (1 + x^k)^2 / (1 - x^(4*k))^2, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
Euler transform of period 4 sequence [2, 0, 2, 2, ...]. - Michael Somos, Mar 02 2019
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EXAMPLE
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a(4)=11 because "(0,4)=(0,3+1)=(0,2+2)=(1,3)=(1,2+1)=(2,2)=(4,0)=(3+1,0)=(2+2,0)=(3,1)=(2+1,1)".
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 44*x^7 + ... - Michael Somos, Mar 02 2019
G.f. = q^-1 + 2*q^3 + 3*q^7 + 6*q^11 + 11*q^15 + 18*q^19 + 28*q^23 + ... - Michael Somos, Mar 02 2019
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MAPLE
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Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q, 2, 100)^2/etaq(q, 1, 100)^2/etaq(q, 4, 100)^2, q, 50):
CoefficientList(sort(convert(GenFun, polynom), q, ascending), q);
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MATHEMATICA
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s = QPochhammer[-1, x]^2/(4*QPochhammer[x^4, x^4]^2) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2, x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Mar 02 2019 *)
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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) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 02 2019 */
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CROSSREFS
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For a version with signs see A274621.
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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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