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A340621 The number of partitions of n without repeated odd parts having more odd parts than even parts.

%I A340621 #22 May 29 2022 13:31:48
%S A340621 0,1,0,1,1,1,2,1,4,2,6,3,9,6,12,10,17,17,22,26,30,40,40,57,55,82,74,
%T A340621 112,103,153,140,203,193,270,262,351,357,458,478,589,641,760,846,971,
%U A340621 1114,1244,1450,1582,1880,2018,2412,2558,3086,3247,3914,4102,4949
%N A340621 The number of partitions of n without repeated odd parts having more odd parts than even parts.
%H A340621 Seiichi Manyama, <a href="/A340621/b340621.txt">Table of n, a(n) for n = 0..10000</a>
%H A340621 B. Kim, E. Kim, and J. Lovejoy, <a href="https://doi.org/10.1016/j.ejc.2020.103159">Parity bias in partitions</a>, European J. Combin., 89 (2020), 103159, 19 pp.
%F A340621 G.f.: Sum_{n>=1} q^(n^2)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2.
%F A340621 a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(3/2) * 5^(3/4) * n). - _Vaclav Kotesovec_, Jan 14 2021
%e A340621 a(8) = 4 counts the partitions [7,1], [5,3], [5,2,1], and [4,3,1].
%p A340621 b:= proc(n, i, c) option remember; `if`(n=0,
%p A340621       `if`(c>0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
%p A340621       `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
%p A340621     end:
%p A340621 a:= n-> b(n$2, 0):
%p A340621 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 13 2021
%t A340621 b[n_, i_, c_] := b[n, i, c] = If[n == 0,
%t A340621      If[c > 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n - i*j, i - 1, c + j*
%t A340621      If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
%t A340621 a[n_] := b[n, n, 0];
%t A340621 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 29 2022, after _Alois P. Heinz_ *)
%o A340621 (PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, sqrt(N), x^(k^2)*(1-x^(2*k))/prod(j=1, k, (1-x^(2*j))^2)))) \\ _Seiichi Manyama_, Jan 14 2021
%Y A340621 Cf. A006950, A143184, A340622, A340623.
%K A340621 nonn
%O A340621 0,7
%A A340621 _Jeremy Lovejoy_, Jan 13 2021