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A058695
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Number of ways to partition 2n+1 into positive integers.
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51
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1, 3, 7, 15, 30, 56, 101, 176, 297, 490, 792, 1255, 1958, 3010, 4565, 6842, 10143, 14883, 21637, 31185, 44583, 63261, 89134, 124754, 173525, 239943, 329931, 451276, 614154, 831820, 1121505, 1505499, 2012558, 2679689, 3554345, 4697205, 6185689, 8118264, 10619863
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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-1 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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Euler transform of period 16 sequence [ 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 1, 3, 1, ...]. - Michael Somos, Apr 25 2003
Expansion of f(x^1, x^7) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
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EXAMPLE
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G.f. = 1 + 3*x + 7*x^2 + 15*x^3 + 30*x^4 + 56*x^5 + 101*x^6 + 176*x^7 + 297*x^8 + ...
G.f. = q^23 + 3*q^71 + 7*q^119 + 15*q^167 + 30*q^215 + 56*q^263 + 101*q^311 + ...
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MAPLE
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a:= n-> combinat[numbpart](2*n+1):
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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]], {i, 1, nn/2}] (* Geoffrey Critzer, Sep 28 2013 *)
(* also *)
(* also *)
Table[Count[IntegerPartitions[3 n - 1], p_ /; MemberQ[p, n]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 2))), 2*n + 1))}; /* Michael Somos, Apr 25 2003 */
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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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