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A035462
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Number of partitions of n into parts 4k-1.
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10
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1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 4, 3, 4, 5, 5, 5, 6, 7, 8, 7, 8, 11, 10, 10, 13, 14, 14, 15, 17, 19, 20, 20, 24, 27, 26, 28, 33, 35, 35, 39, 44, 46, 48, 52, 58, 62, 63, 69, 78, 80, 83, 93, 100, 104, 111, 120, 130, 137, 143, 156, 169, 175, 185, 203
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OFFSET
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0,15
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COMMENTS
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Also, number of partitions into parts 8k+3 or 8k+7.
Also number of partitions of n such that 2k-1 and 2k occur with the same multiplicity. Example: a(18)=3 because we have [8,7,2,1],[6,5,4,3] and [2,2,2,2,2,2,1,1,1,1,1,1]. It is easy to find a bijection between these partitions and those described in the definition. - Emeric Deutsch, Apr 05 2006
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} (x^(3*n) / Product_{k=1..n} (1 - x^(4*k))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k>=n} (1 - x^(4*k+3))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k=0..n} (1 - x^(4*k+3))). - Joerg Arndt, Apr 08 2011
a(n) ~ Pi^(3/4) * exp(Pi*sqrt(n/6)) / (Gamma(1/4) * 2^(13/8) * 3^(3/8) * n^(7/8)) * (1 + (Pi/(96*sqrt(6)) - 21*sqrt(3/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
G.f.: A(x) = Sum_{n >= 0} x^(n*(4*n-1))/Product_{k = 1..n} ( (1 - x^(4*k))*(1 - x^(4*k-1)) ). (Set z = x^3 and q = x^4 in Mc Laughlin et al., Section 1.3, Entry 7.)
Similarly, A(x) = Sum_{n >= 0} x^(n*(4*n+3))/( (1 - x^3)*Product_{k = 1..n} ((1 - x^(4*k))*(1 - x^(4*k+3))) ). (End)
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EXAMPLE
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a(18)=3 because we have [15,3],[11,7] and [3,3,3,3,3,3].
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MAPLE
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g:=1/product(1-x^(4*i-1), i=1..50): gser:=series(g, x=0, 80): seq(coeff(gser, x, n), n=1..75); # Emeric Deutsch, Apr 05 2006
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+3)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 - 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 04 2020 *)
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CROSSREFS
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Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), this sequence (m=4), A109700 (m=5), A109702 (m=6), A109708 (m=7).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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