A340622 -id:A340622 - cp's OEIS frontend

A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

0, 0, 1, 0, 2, 1, 3, 3, 5, 7, 8, 13, 14, 23, 23, 37, 39, 59, 63, 90, 101, 136, 156, 201, 239, 296, 355, 428, 523, 617, 754, 878, 1078, 1243, 1517, 1741, 2121, 2426, 2928, 3348, 4021, 4596, 5468, 6257, 7400, 8472, 9936, 11389, 13285, 15233, 17645, 20244, 23346

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(7) = 3 counts the partitions [4,2,1], [3,2,2], and [2,2,2,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A340621, A340622, A340647.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c<0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k - 1))/(1 - x^(2*k)), {k, 1, nmax/2}] - Sum[x^(k^2)/Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(prod(k=1, N, (1+x^(2*k-1))/(1-x^(2*k)))-sum(k=0, sqrt(N), x^(k^2)/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021

G.f.: (Product_{k>=1} (1+q^(2*k-1))/(1-q^(2*k))) - Sum_{n>=0} q^(n^2)/Product_{k=1..n} (1-q^(2*k))^2.

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

Original entry on oeis.org

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, 112, 103, 153, 140, 203, 193, 270, 262, 351, 357, 458, 478, 589, 641, 760, 846, 971, 1114, 1244, 1450, 1582, 1880, 2018, 2412, 2558, 3086, 3247, 3914, 4102, 4949

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(8) = 4 counts the partitions [7,1], [5,3], [5,2,1], and [4,3,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A143184, A340622, A340623.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c>0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c > 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n - i*j, i - 1, c + j*
     If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

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

G.f.: Sum_{n>=1} q^(n^2)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2.

a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(3/2) * 5^(3/4) * n). - Vaclav Kotesovec, Jan 14 2021

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A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

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