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A100847
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Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.
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3
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1, 2, 3, 7, 10, 17, 28, 42, 62, 93, 137, 193, 276, 383, 532, 734, 997, 1342, 1807, 2400, 3177, 4190, 5478, 7130, 9245, 11923, 15305, 19591, 24957, 31673, 40075, 50518, 63460, 79523, 99296, 123664, 153616, 190271, 235072, 289776, 356302, 437107, 535112, 653626
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^i).
a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((2*Pi^2/3 + 8*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016
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EXAMPLE
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a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.
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MAPLE
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g:=product((1+x^i-x^(2*i))/(1-x^i), i=1..50): gser:=series(g, x=0, 40): seq(coeff(gser, x, n), n=0..35); # Emeric Deutsch, Aug 25 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(irem(i+j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
+b(n, i-1)))
end:
a:= n-> b(2*n$2):
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(2*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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