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A100824
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Number of partitions of n with at most one odd part.
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10
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1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742
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OFFSET
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0,4
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COMMENTS
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Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:
1 11 21 22 32 33 43 44 54
111 1111 221 2211 331 2222 441
2111 111111 2221 3311 3222
11111 3211 221111 3321
22111 11111111 4311
211111 22221
1111111 33111
222111
321111
2211111
21111111
111111111
(End)
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LINKS
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FORMULA
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G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).
a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016
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EXAMPLE
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The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (22) (32) (42) (43) (44) (54)
(41) (222) (52) (62) (63)
(221) (61) (422) (72)
(322) (2222) (81)
(421) (432)
(2221) (441)
(522)
(621)
(3222)
(4221)
(22221)
(End)
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MAPLE
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seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i), i=1..100), x, 100), polynom), x, n), n=0..60); (C. Ronaldo)
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MATHEMATICA
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nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n], Count[#, _?OddQ]<=1&]], {n, 0, 30}] (* Gus Wiseman, Jan 21 2022 *)
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PROG
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(PARI) a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022
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CROSSREFS
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The case of alternating sum 0 (equality) is A000070.
A multiplicative version is A339846.
A058695 = partitions of odd numbers.
A277103 = partitions with the same number of odd parts as their conjugate.
Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.
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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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More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
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STATUS
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approved
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