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A341497 Number of partitions of n with exactly one repeated part and that part is odd. 5

%I

%S 0,0,1,1,2,3,5,7,9,13,17,23,30,39,49,63,78,98,122,150,184,225,272,329,

%T 397,475,567,676,802,948,1121,1317,1545,1810,2112,2460,2863,3319,3842,

%U 4442,5123,5897,6782,7780,8913,10200,11648,13285,15136,17214,19555,22191,25143

%N Number of partitions of n with exactly one repeated part and that part is odd.

%H Andrew Howroyd, <a href="/A341497/b341497.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.: (Sum_{k>=1} x^(4*k-2)/(1 - x^(4*k-2)) * Product_{k>=1} (1 + x^k).

%F a(n) = A090867(n) - A341496(n).

%F a(n) = A116680(n) + A341496(n).

%F a(n) = A341495(n) for even n; a(n) = A341494(n) for odd n.

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

%e The a(3) = 1 partition is: 1+1+1.

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

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

%o (PARI) seq(n)={Vec(sum(k=1, (n+2)\4, x^(4*k-2)/(1 - x^(4*k-2)) + O(x*x^n)) * prod(k=1, n, 1 + x^k + O(x*x^n)), -(n+1))}

%Y Cf. A090867, A116680, A341494, A341495, A341496.

%K nonn

%O 0,5

%A _Andrew Howroyd_, Feb 13 2021

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Last modified February 6 18:02 EST 2023. Contains 360111 sequences. (Running on oeis4.)