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A341495 Number of partitions of n into an odd number of parts such that the set of even parts has only one element. 5

%I

%S 0,0,1,0,2,1,5,2,9,5,17,9,30,16,49,26,78,43,122,67,184,101,272,151,

%T 397,222,567,320,802,454,1121,637,1545,884,2112,1214,2863,1651,3842,

%U 2227,5123,2979,6782,3957,8913,5218,11648,6840,15136,8914,19555,11552,25143

%N Number of partitions of n into an odd number of parts such that the set of even parts has only one element.

%H Andrew Howroyd, <a href="/A341495/b341495.txt">Table of n, a(n) for n = 0..1000</a>

%H Cristina Ballantine and Mircea Merca, <a href="https://doi.org/10.1007/s11139-019-00184-7">Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts</a>, Ramanujan J., 54:1 (2021), 107-112.

%F 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)).

%F a(n) = A090867(n) - A341494(n).

%e The a(2) = 1 partition is: 2.

%e The a(4) = 2 partitions are: 4, 1+1+2.

%e The a(5) = 1 partition is: 1+2+2.

%e The a(6) = 5 partitions are: 6, 1+1+4, 1+2+3, 2+2+2, 1+1+1+1+2.

%t P[n_, c_] := c*Sum[x^(2k)/(1 - c*x^(2k)) + O[x]^n, {k, 1, n/2}]/

%t Product[1 - c*x^(2k - 1) + O[x]^n, {k, 1, n/2}];

%t CoefficientList[(P[100, 1] - P[100, -1])/2, x] (* _Jean-Fran├žois Alcover_, May 24 2021, from PARI code *)

%o (PARI)

%o 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))}

%o seq(n)={Vec(P(n,1) - P(n,-1), -(n+1))/2}

%Y Cf. A090867, A341494, A341496, A341497.

%K nonn

%O 0,5

%A _Andrew Howroyd_, Feb 13 2021

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Last modified March 29 03:40 EDT 2023. Contains 361596 sequences. (Running on oeis4.)