A067661 -id:A067661 - cp's OEIS frontend

A339382 Number of partitions of n into an even number of distinct primes (counting 1 as a prime).

1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 6, 5, 5, 5, 6, 6, 7, 7, 9, 8, 9, 8, 11, 10, 11, 12, 14, 12, 15, 14, 17, 16, 17, 17, 22, 20, 22, 21, 25, 24, 28, 27, 31, 30, 33, 31, 39, 36, 40, 40, 46, 42, 49, 47, 54, 53, 58, 55, 67, 63, 70, 68

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].

Index entries for sequences related to partitions

Cf. A000586, A008578, A036497, A067661, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339383.

s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
      `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
       b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 02 2020

nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] + (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * ((1 + x) * Product_{k>=1} (1 + x^prime(k)) + (1 - x) * Product_{k>=1} (1 - x^prime(k))).

a(n) = (A036497(n) + A298602(n)) / 2.

A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k(2k-1)) / Product_{j=1..2*k-1} (1 - x^j).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 55, 69, 85, 104, 127, 154, 186, 224, 268, 320, 381, 452, 534, 630, 741, 869, 1017, 1187, 1382, 1606, 1862, 2155, 2489, 2869, 3301, 3792, 4349, 4979, 5692, 6497, 7405, 8429, 9581, 10876, 12331, 13963, 15792, 17840, 20131, 22691

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

Partial sums of A067659.

			From _Gus Wiseman_, Jul 18 2021: (Start)
Also the number of strict integer partitions of 2n+1 of even length with exactly one odd part. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (2,1)  (3,2)  (4,3)  (5,4)  (6,5)   (7,6)      (8,7)      (9,8)
         (4,1)  (5,2)  (6,3)  (7,4)   (8,5)      (9,6)      (10,7)
                (6,1)  (7,2)  (8,3)   (9,4)      (10,5)     (11,6)
                       (8,1)  (9,2)   (10,3)     (11,4)     (12,5)
                              (10,1)  (11,2)     (12,3)     (13,4)
                                      (12,1)     (13,2)     (14,3)
                                      (6,4,2,1)  (14,1)     (15,2)
                                                 (6,4,3,2)  (16,1)
                                                 (8,4,2,1)  (6,5,4,2)
                                                            (8,4,3,2)
                                                            (8,6,2,1)
                                                            (10,4,2,1)
Also the number of integer partitions of 2n+1 covering an initial interval and having even maximum and alternating sum 1.
(End)

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.2 "Partitions into distinct parts", page 350.
Index entries for sequences related to partitions

Partial sums of A067659.
The following relate to strict integer partitions of 2n+1 of even length with exactly one odd part.
- Allowing any length gives A036469.
- The non-strict version is A306145.
- The version for odd length is A318155 (non-strict: A304620).
- Allowing any number of odd parts gives A343942 (odd bisection of A067661).
A000041 counts partitions.
A027187 counts partitions of even length (strict: A067661).
A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
Cf. A000070, A030229, A035294, A058696, A078616, A087447, A152146, A236559, A343941, A344611, A344739.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);

nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k - 1))/Product[(1 - x^j), {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] - QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman, Jul 18 2021 *)

a(n) = A036469(n) - A318155(n).
a(n) = A318155(n) - A078616(n).
a(n) ~ exp(Pi*sqrt(n/3)) * 3^(1/4) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 20 2018

A238208 The total number of 1's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 24, 28, 33, 38, 45, 52, 60, 69, 80, 91, 105, 120, 137, 156, 178, 202, 230, 261, 295, 334, 378, 426, 481, 542, 609, 685, 769, 862, 966, 1082, 1209, 1351, 1508, 1681, 1873, 2086, 2319, 2578

Offset: 0

Mircea Merca, Feb 20 2014

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

Or: the number of partitions of n-1 into an even number of distinct parts >=2. - R. J. Mathar, May 11 2016

			a(10) = 3 because the partitions in question are: 7+2+1, 6+3+1, 5+4+1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Andrew Howroyd)

Column k=1 of A238450.

Cf. A067659, A067661, A133280.

A238208 := proc(n)
    local a,L,Lset;
    a := 0 ;
    L := combinat[firstpart](n) ;
    while true do
        # check that parts are distinct
        Lset := convert(L,set) ;
        if nops(L) = nops(Lset) then
            # check that number is odd
            if type(nops(L),'odd') then
                if 1 in Lset then
                    a := a+1 ;
                end if;
            end if;
        end if;
        L := combinat[nextpart](L) ;
        if L = FAIL then
            return a;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 11 2016
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))
    end:
a:= n-> b(n-1, 2, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, May 01 2020

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i+1, t] + b[n-i, i+1, 1-t]]];
a[n_] := b[n-1, 2, 1];
a /@ Range[0, 100] (* Jean-François Alcover, May 17 2020, after Alois P. Heinz *)

seq(n)={my(A=O(x^n)); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x)) + eta(x + A)/(1-x))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/2)} A067661(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067659(n-2*j).
G.f.: (1/2)*(x/(1+x))*(Product_{n>=1} 1 + x^n) + (1/2)*(x/(1-x))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 17 2020
From Peter Bala, Feb 02 2021: (Start)
a(n+1) = d(n) - ( d(n-1) + d(n-3) ) + ( d(n-4) + d(n-6) + d(n-8) ) - ( d(n-9) + d(n-11) + d(n-13) + d(n-15) ) + ( d(n-16) + d(n-18) + d(n-20) + d(n-22) + d(n-24) ) - ( d(n-25) + d(n-27) + d(n-29) + d(n-31) + d(n-33) + d(n-35) ) + ..., where d(n) = A000009(n) is the number of partitions of n into distinct parts, with the convention that d(n) = 0 for n < 0.
G.f.: x/(1 - x^2)*Sum_{n >= 0} (-1)^n*x^((n^2+n+1-(-1)^n)/2)/Product_{k = 1..n} 1 - x^k.
Alternative g.f.: ( Product_{k >= 1} 1 + x^k ) * x*Sum_{n >= 0} (-1)^n*x^(n^2)*(1 - x^(2*n+2))/(1 - x^2).
Faster converging g.f. (conjecture): Sum_{n >= 0} x^((n+1)*(2*n+1))/ Product_{k = 1..2*n} 1 - x^k. (End)

a(51)-a(60) from R. J. Mathar, May 11 2016

A238209 The total number of 2's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 13, 16, 18, 22, 26, 30, 35, 41, 48, 55, 64, 73, 85, 97, 111, 127, 146, 165, 189, 214, 244, 276, 313, 353, 400, 451, 508, 572, 644, 722, 811, 909, 1018, 1139, 1273, 1421, 1586, 1768, 1968, 2191, 2436

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(11) = 3 because the partitions in question are: 8+2+1, 6+3+2, 5+4+2.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=2 of A238450.

Cf. A067659, A067661.

nmax = 100; With[{k=2}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

seq(n)={my(A=O(x^(n-1))); Vec(x^2*(eta(x^2 + A)/(eta(x + A)*(1+x^2)) + eta(x + A)/(1-x^2))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/4)} A067661(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067659(n-4*j).
G.f.: (1/2)*(x^2/(1+x^2))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^2/(1-x^2))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

A238210 The total number of 3's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 23, 28, 32, 37, 44, 51, 58, 68, 78, 89, 103, 118, 134, 154, 175, 199, 227, 257, 291, 330, 373, 421, 475, 535, 602, 677, 760, 852, 955, 1069, 1196, 1336, 1491, 1663, 1853, 2063, 2296

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 3 because the partitions in question are: 8+3+1, 7+3+2, 5+4+3.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=3 of A238450.

Cf. A067659, A067661.

nmax = 100; With[{k=3}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

seq(n)={my(A=O(x^(n-2))); Vec(x^3*(eta(x^2 + A)/(eta(x + A)*(1+x^3)) + eta(x + A)/(1-x^3))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/6)} A067661(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067659(n-6*j).
G.f.: (1/2)*(x^3/(1+x^3))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^3/(1-x^3))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

A238211 The total number of 4's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 5, 5, 6, 7, 9, 11, 13, 15, 18, 21, 25, 29, 34, 40, 46, 54, 62, 71, 82, 95, 108, 124, 142, 162, 184, 210, 238, 271, 306, 346, 392, 443, 498, 561, 632, 710, 796, 893, 1000, 1120, 1252, 1397, 1560, 1740, 1937, 2156

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 3 because the partitions in question are: 7+4+1, 6+4+2, 5+4+3.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=4 of A238450.

Cf. A067659, A067661.

nmax = 100; With[{k=4}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/8)} A067661(n-(2*j-1)*4) - Sum_{j=1..floor(n/8)} A067659(n-8*j).
G.f.: (1/2)*(x^4/(1+x^4))*(Product{n>=1} 1 + x^n) + (1/2)*(x^4/(1-x^4))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

A238212 The total number of 5's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 1, 2, 2, 3, 5, 4, 5, 7, 8, 10, 11, 13, 16, 19, 23, 26, 31, 36, 42, 49, 56, 65, 75, 86, 100, 114, 130, 149, 170, 193, 220, 250, 283, 321, 363, 410, 463, 522, 587, 660, 742, 832, 933, 1045, 1168, 1307, 1459, 1627, 1814, 2020

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 2 because the partitions in question are: 6+5+1, 5+4+3.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=5 of A238450.

Cf. A067659, A067661.

tn5[n_]:=Module[{op=IntegerPartitions[n],m},m=Flatten[Select[op,OddQ[ Length[#]] && Length[#]==Length[Union[#]]&]];Count[m,5]]; Array[tn5,60,0] (* Harvey P. Dale, Feb 06 2015 *)
nmax = 100; With[{k=5}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/10)} A067661(n-(2*j-1)*5) - Sum_{j=1..floor(n/10)} A067659(n-10*j).
G.f.: (1/2)*(x^5/(1+x^5))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^5/(1-x^5))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

A238213 The total number of 6's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 2, 2, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 17, 20, 23, 27, 33, 38, 44, 51, 59, 68, 79, 91, 104, 119, 136, 155, 178, 202, 230, 261, 296, 335, 379, 428, 483, 544, 612, 688, 773, 867, 972, 1088, 1217, 1360, 1518, 1693, 1887

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 2 because the partitions in question are: 6+5+1, 6+4+2.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=6 of A238450.

Cf. A067659, A067661.

nmax = 100; With[{k=6}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/12)} A067661(n-(2*j-1)*6) - Sum_{j=1..floor(n/12)} A067659(n-12*j).
G.f.: (1/2)*(x^6/(1+x^6))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^6/(1-x^6))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

A238215 The total number of 1's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18, 21, 24, 28, 33, 38, 44, 51, 59, 68, 79, 90, 104, 119, 136, 156, 178, 202, 230, 261, 296, 335, 379, 427, 482, 543, 610, 686, 770, 863, 967, 1082, 1209, 1351, 1508, 1681, 1873, 2085, 2318, 2577

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 3 because the partitions in question are: 11+1, 6+3+2+1, 5+4+2+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=1 of A238451.

Cf. A067659, A067661.

b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))
    end:
a:= n-> b(n-1, 2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, May 01 2020

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 1, 1 - t]]];
a[n_] := b[n - 1, 2, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

seq(n)={my(A=O(x^n)); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x)) - eta(x + A)/(1-x))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/2)} A067659(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067661(n-2*j).
G.f.: (1/2)*(x/(1+x))*(Product_{n>=1} 1 + x^n) - (1/2)*(x/(1-x))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Nov 07 2024

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

A238217 The total number of 2's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 11, 14, 16, 19, 22, 26, 30, 35, 41, 47, 55, 63, 73, 84, 97, 110, 127, 145, 166, 188, 215, 243, 277, 313, 354, 400, 452, 508, 573, 644, 723, 811, 910, 1018, 1139, 1273, 1421, 1586, 1768, 1968, 2190, 2436

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 3 because the partitions in question are: 10+2, 6+3+2+1, 5+4+2+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=2 of A238451.

Cf. A067659, A067661.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Length[#] == Length[ Union[#]]&&MemberQ[#,2]&]],{n,0,50}] (* Harvey P. Dale, Dec 09 2014 *)
nmax = 100; With[{k=2}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

seq(n)={my(A=O(x^(n-1))); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x^2)) - eta(x + A)/(1-x^2))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/4)} A067659(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067661(n-4*j).
G.f.: (1/2)*(x^2/(1+x^2))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^2/(1-x^2))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

cp's OEIS Frontend

A339382 Number of partitions of n into an even number of distinct primes (counting 1 as a prime).

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A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).

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A238208 The total number of 1's in all partitions of n into an odd number of distinct parts.

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A238209 The total number of 2's in all partitions of n into an odd number of distinct parts.

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A238210 The total number of 3's in all partitions of n into an odd number of distinct parts.

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A238211 The total number of 4's in all partitions of n into an odd number of distinct parts.

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A238212 The total number of 5's in all partitions of n into an odd number of distinct parts.

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A318156 Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k(2k-1)) / Product_{j=1..2*k-1} (1 - x^j).