cp's OEIS Frontend

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

%I A341497 #21 Jan 15 2025 04:17:34
%S A341497 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 A341497 397,475,567,676,802,948,1121,1317,1545,1810,2112,2460,2863,3319,3842,
%U A341497 4442,5123,5897,6782,7780,8913,10200,11648,13285,15136,17214,19555,22191,25143
%N A341497 Number of partitions of n with exactly one repeated part and that part is odd.
%H A341497 Andrew Howroyd, <a href="/A341497/b341497.txt">Table of n, a(n) for n = 0..1000</a>
%H A341497 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 A341497 G.f.: (Sum_{k>=1} x^(4*k-2)/(1 - x^(4*k-2))) * Product_{k>=1} (1 + x^k).
%F A341497 a(n) = A090867(n) - A341496(n).
%F A341497 a(n) = A116680(n) + A341496(n).
%F A341497 a(n) = A341495(n) for even n; a(n) = A341494(n) for odd n.
%F A341497 a(n) = (A067588(n) - A116676(n))/2. - _Peter Bala_, Jan 13 2025
%e A341497 The a(2) = 1 partition is: 1+1.
%e A341497 The a(3) = 1 partition is: 1+1+1.
%e A341497 The a(4) = 2 partitions are: 1+1+2, 1+1+1+1.
%e A341497 The a(5) = 3 partitions are: 1+1+3, 1+1+1+2, 1+1+1+1+1.
%o A341497 (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 A341497 Cf. A067588, A090867, A116676, A116680, A341494, A341495, A341496.
%K A341497 nonn,easy
%O A341497 0,5
%A A341497 _Andrew Howroyd_, Feb 13 2021