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A058696
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Number of ways to partition 2n into positive integers.
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84
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1, 2, 5, 11, 22, 42, 77, 135, 231, 385, 627, 1002, 1575, 2436, 3718, 5604, 8349, 12310, 17977, 26015, 37338, 53174, 75175, 105558, 147273, 204226, 281589, 386155, 526823, 715220, 966467, 1300156, 1741630, 2323520, 3087735, 4087968, 5392783, 7089500, 9289091
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of partitions of 3n-2 having n as a part, for n >= 1. Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1. - Clark Kimberling, Mar 02 2014
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LINKS
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FORMULA
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Expansion of f(x^3, x^5) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
Euler transform of period 16 sequence [ 2, 2, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, ...]. - Michael Somos, Apr 25 2003
G.f.: Product_{k>=1} (1 + x^(8*k-4)) * (1 + x^(8*k)) * (1 + x^k)^2 / ((1 + x^(8*k-1)) * (1 + x^(8*k-7)) * (1 - x^k)). - Vaclav Kotesovec, Nov 17 2016
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EXAMPLE
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G.f. = 1 + 2*x + 5*x^2 + 11*x^3 + 22*x^4 + 42*x^5 + 77*x^6 + 135*x^7 + ...
G.f. = q^-1 + 2*q^47 + 5*q^95 + 11*q^143 + 22*q^191 + 42*q^239 + 77*q^287 + ...
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MAPLE
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a:= n-> combinat[numbpart](2*n):
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MATHEMATICA
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nn=100; Table[CoefficientList[Series[Product[1/(1-x^i), {i, 1, nn}], {x, 0, nn}], x][[2i-1]], {i, 1, nn/2}] (* Geoffrey Critzer, Sep 28 2013 *)
(* also *)
(* also *)
Table[Count[IntegerPartitions[3 n - 2], p_ /; MemberQ[p, n]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(8*k-4))*(1 + x^(8*k))*(1 + x^k)^2/((1 + x^(8*k-1))*(1 + x^(8*k-7))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2016 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 1))), 2*n))}; /* Michael Somos, Apr 25 2003 */
(MuPAD) combinat::partitions::count(2*i) $i=0..54 // Zerinvary Lajos, Apr 16 2007
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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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