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A341495
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Number of partitions of n into an odd number of parts such that the set of even parts has only one element.
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5
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0, 0, 1, 0, 2, 1, 5, 2, 9, 5, 17, 9, 30, 16, 49, 26, 78, 43, 122, 67, 184, 101, 272, 151, 397, 222, 567, 320, 802, 454, 1121, 637, 1545, 884, 2112, 1214, 2863, 1651, 3842, 2227, 5123, 2979, 6782, 3957, 8913, 5218, 11648, 6840, 15136, 8914, 19555, 11552, 25143
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OFFSET
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0,5
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.
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FORMULA
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G.f.: (P(x,1) - P(x,-1))/2 where P(x,c) = (Sum_{k>=1} c*x^(2*k)/(1-c*x^(2*k))) / (Product_{k>=1} 1-c*x^(2*k-1)).
a(n) = A090867(n) - A341494(n).
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EXAMPLE
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The a(2) = 1 partition is: 2.
The a(4) = 2 partitions are: 4, 1+1+2.
The a(5) = 1 partition is: 1+2+2.
The a(6) = 5 partitions are: 6, 1+1+4, 1+2+3, 2+2+2, 1+1+1+1+2.
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MATHEMATICA
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P[n_, c_] := c*Sum[x^(2k)/(1 - c*x^(2k)) + O[x]^n, {k, 1, n/2}]/
Product[1 - c*x^(2k - 1) + O[x]^n, {k, 1, n/2}];
CoefficientList[(P[100, 1] - P[100, -1])/2, x] (* Jean-François Alcover, May 24 2021, from PARI code *)
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PROG
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(PARI)
P(n, c)={c*sum(k=1, n\2, x^(2*k)/(1-c*x^(2*k)) + O(x*x^n))/prod(k=1, n\2, 1-c*x^(2*k-1) + O(x*x^n))}
seq(n)={Vec(P(n, 1) - P(n, -1), -(n+1))/2}
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CROSSREFS
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Cf. A090867, A341494, A341496, A341497.
Sequence in context: A205377 A082010 A318972 * A337943 A275213 A113176
Adjacent sequences: A341492 A341493 A341494 * A341496 A341497 A341498
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KEYWORD
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nonn
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AUTHOR
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Andrew Howroyd, Feb 13 2021
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STATUS
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approved
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