A236913 -id:A236913 - cp's OEIS frontend

A027187 Number of partitions of n into an even number of parts.

1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338, 73680, 86716, 102162, 119918

Offset: 0

Clark Kimberling

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]
Number of partitions of n+1 into an odd number of parts, the least being 1.
Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...
From _Gus Wiseman_, Jan 05 2021: (Start)
The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (2211)    (2221)    (71)
                              (3111)    (3211)    (2222)
                              (111111)  (4111)    (3221)
                                        (211111)  (3311)
                                                  (4211)
                                                  (5111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)
The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.
  (2)  (21)  (4)    (41)    (6)      (43)      (8)
             (22)   (221)   (42)     (61)      (44)
             (211)  (2111)  (222)    (421)     (62)
                            (411)    (2221)    (422)
                            (2211)   (4111)    (431)
                            (21111)  (22111)   (611)
                                     (211111)  (2222)
                                               (4211)
                                               (22211)
                                               (41111)
                                               (221111)
                                               (2111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 13.
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000701, A046682, A026838, A102462.
The Heinz numbers of these partitions are A028260.
The odd version is A027193.
The strict case is A067661.
The case of even sum as well as length is A236913 (the even bisection).
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A000009 counts partitions into odd parts, ranked by A066208.
A026805 counts partitions whose least part is even.
A072233 counts partitions by sum and length.
A101708 counts partitions of even positive rank.
Cf. A000700, A026424, A058696, A096373, A244990, A300061.

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)
a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

my(q='q+O('q^66)); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.
a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004
From Bill Gosper, Jun 25 2005: (Start)
G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3*q^4 + 3*q^5 + 6*q^6 + ...
  = Sum_{n >= 0} q^(2*n)/(q; q)_{2*n}
  = ((Product_{k >= 1} 1/(1-q^k)) + (Product_{k >= 1} 1/(1+q^k)))/2.
Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + ...
Then B(q) = Sum_{n >= 0} q^(2*n+1)/(q; q){2*n+1} = ((Product{k >= 1} 1/(1-q^k)) - (Product_{k >= 1} 1/(1+q^k)))/2.
Also we have the following identity involving 2 X 2 matrices:
Product_{k >= 1} [ 1/(1-q^(2*k)), q^k/(1-q^(2*k)) ; q^k/(1-q^(2*k)), 1/(1-q^(2*k)) ]
  = [ A(q), B(q) ; B(q), A(q) ]. (End)
a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006
Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006
G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006
a(n) = A338914(n) + A096373(n). - Gus Wiseman, Jan 06 2021

Offset changed to 0 by Michael Somos, Jul 24 2012

A035363 Number of partitions of n into even parts.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
  1  .  3   .  5    .  7     .  9      .  B       .  D
        21     41      43       63        65         85
               221     61       81        83         A3
                       421      441       A1         C1
                       2221     621       443        643
                                4221      641        661
                                22221     821        841
                                          4421       A21
                                          6221       4441
                                          42221      6421
                                          222221     8221
                                                     44221
                                                     62221
                                                     422221
                                                     2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
  (11)  (22)   (33)    (44)     (55)      (66)
        (211)  (321)   (422)    (532)     (633)
               (3111)  (431)    (541)     (642)
                       (4211)   (5221)    (651)
                       (41111)  (5311)    (6222)
                                (52111)   (6321)
                                (511111)  (6411)
                                          (62211)
                                          (63111)
                                          (621111)
                                          (6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
  (2)  (22)  (222)  (2222)  (22222)  (222222)  (2222222)
       (31)  (321)  (3221)  (32221)  (322221)  (3222221)
             (411)  (3311)  (33211)  (332211)  (3322211)
                    (4211)  (42211)  (333111)  (3332111)
                    (5111)  (43111)  (422211)  (4222211)
                            (52111)  (432111)  (4322111)
                            (61111)  (441111)  (4331111)
                                     (522111)  (4421111)
                                     (531111)  (5222111)
                                     (621111)  (5321111)
                                     (711111)  (5411111)
                                               (6221111)
                                               (6311111)
                                               (7211111)
                                               (8111111)
(End)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

A339846 Number of even-length factorizations of n into factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 5, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 1, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 6, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 3, 1, 0, 6, 1, 1, 1, 4, 0, 6, 1, 2, 1, 1, 1, 10, 0, 2, 2, 5, 0, 3, 0, 4, 3

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 12, 16, 24, 36, 48, 72, 96, 120:
  2*6  2*8      3*8      4*9      6*8      8*9      2*48         2*60
  3*4  4*4      4*6      6*6      2*24     2*36     3*32         3*40
       2*2*2*2  2*12     2*18     3*16     3*24     4*24         4*30
                2*2*2*3  3*12     4*12     4*18     6*16         5*24
                         2*2*3*3  2*2*2*6  6*12     8*12         6*20
                                  2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                           2*2*3*6  2*2*4*6      10*12
                                           2*3*3*4  2*3*4*4      2*2*5*6
                                                    2*2*2*12     2*3*4*5
                                                    2*2*2*2*2*3  2*2*2*15
                                                                 2*2*3*10

The case of set partitions (or n squarefree) is A024430.
The case of partitions (or prime powers) is A027187.
The ordered version is A174725, odd: A174726.
The odd-length factorizations are counted by A339890.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A316439 counts factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Cf. A002033, A007716, A027193, A050320, A058695, A074206, A236913, A320655, A320656, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, g(n$2, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],EvenQ@Length[#]&]],{n,100}]

A339846(n, m=n, e=1) = if(1==n, e, sumdiv(n, d, if((d>1)&&(d<=m), A339846(n/d, d, 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(n) + A339890(n) = A001055(n).

Data section extended up to a(105) by Antti Karttunen, Oct 22 2023

A067661 Number of partitions of n into distinct parts such that number of parts is even.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3959, 4404

Offset: 0

Naohiro Nomoto, Feb 23 2002

Ramanujan theta functions: phi(q) (A000122), chi(q) (A000700).

			G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(3) = 1 through a(14) = 11 partitions (A-D = 10..13):
  21   31   32   42   43   53   54   64     65     75     76     86
            41   51   52   62   63   73     74     84     85     95
                      61   71   72   82     83     93     94     A4
                                81   91     92     A2     A3     B3
                                     4321   A1     B1     B2     C2
                                            5321   5421   C1     D1
                                                   6321   5431   5432
                                                          6421   6431
                                                          7321   6521
                                                                 7421
                                                                 8321
(End)

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (2).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Dominates A000009.
Numbers with these strict partitions as binary indices are A001969.
The non-strict case is A027187, ranked by A028260.
The Heinz numbers of these partitions are A030229.
The odd version is A067659, ranked by A030059.
The version for rank is A117192, with positive case A101708.
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A008289 counts strict partitions by sum and length.
A026805 counts partitions whose least part is even.
Cf. A000700, A027193, A058696, A072233, A096373, A236913, A244990, A300061.

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:= n-> b(n$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] + QPochhammer[ x]) / 2, {x, 0, n}]; (* Michael Somos, May 06 2015 *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 08 2021 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A) + eta(x + A)) / 2, n))}; /* Michael Somos, Feb 14 2006 */

N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);
gf=sum(n=0, S, (n%2==0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
Vec(gf)  \\ Joerg Arndt, Apr 01 2014

G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^3 + q^4 + 2 q^5 + 2 q^6 + 3 q^7 + ... = Sum_{n >= 0} q^(n(2n+1))/(q; q){2n} [_Bill Gosper, Jun 25 2005]
Also, let B(q) = Sum_{n >= 0} A067659(n) q^n = q + q^2 + q^3 + q^4 + q^5 + 2 q^6 + ... Then B(q) = Sum_{n >= 0} q^((n+1)(2n+1))/(q; q)_{2n+1}.
Also we have the following identity involving 2 X 2 matrices:
Prod_{k >= 1} [ 1, q^k; q^k, 1 ] = [ A(q), B(q); B(q), A(q) ] [Bill Gosper, Jun 25 2005]
a(n) = (A000009(n)+A010815(n))/2. - Vladeta Jovovic, Feb 24 2002
Expansion of (1 + phi(-x)) / (2*chi(-x)) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006
a(n) + A067659(n) = A000009(n). - R. J. Mathar, Jun 18 2016
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 24 2018
A000009(n) = a(n) + A067659(n). - Gus Wiseman, Jan 09 2021
From Peter Bala, Feb 05 2021: (Start)
G.f.: A(x) = (1/2)*((Product_{n >= 0} 1 + x^n) + (Product_{n >= 0} 1 - x^n)).
Let B(x) denote the g.f. of A067659. Then
A(x)^2 - B(x)^2 = A(x^2) - B(x^2) = Product_{n >= 1} 1 - x^(2*n) = Sum_{n in Z} (-1)^n*x^(n*(3*n+1)).
A(x) + B(x) is the g.f. of A000009.
1/(A(x) - B(x)) is the g.f. of A000041.
(A(x) + B(x))/(A(x) - B(x)) is the g.f. of A015128.
A(x)/(A(x) + B(x)) = Sum_{n >= 0} (-1)^n*x^n^2 = (1 + theta_3(-x))/2.
B(x)/(A(x) - B(x)) is the g.f. of A014968.
A(x)/(A(x^2) - B(x^2)) is the g.f. of A027187.
B(x)/(A(x^2) - B(x^2)) is the g.f. of A027193. (End)

A236559 Number of partitions of 2n of type EO (see Comments).

Original entry on oeis.org

0, 1, 2, 5, 10, 20, 37, 66, 113, 190, 310, 497, 782, 1212, 1851, 2793, 4163, 6142, 8972, 12989, 18646, 26561, 37556, 52743, 73593, 102064, 140736, 193011, 263333, 357521, 483129, 649960, 870677, 1161604, 1543687, 2043780, 2696156, 3544485, 4644241, 6065739

Offset: 0

Clark Kimberling, Feb 01 2014

The partitions of n are partitioned into four types:
EO, even # of odd parts and odd  # of even parts, A236559;
OE, odd  # of odd parts and even # of even parts, A160786;
EE, even # of odd parts and even # of even parts, A236913;
OO, odd  # of odd parts and odd  # of even parts, A236914.
A236559 and A160786 are the bisections of A027193;
A236913 and A236914 are the bisections of A027187.

			The partitions of 4 of type EO are [4] and [2,1,1], so that a(2) = 2.
type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11
EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0
OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29
EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0
OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000041, A000701, A027187, A027193, A160786, A236913, A236914.

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
      `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
      `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
          [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
    end:
a:= n-> b(2*n$2)[3]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 16 2014

z = 25; m1 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,  OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]]; m2 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,       OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]]; m3 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];
m1 (* A236559, type EO*)
m2 (* A160786, type OE*)
m3 (* A236913, type EE*)
m4 (* A236914, type OO*)
(* Peter J. C. Moses, Feb 03 2014 *)
b[n_, i_] := b[n, i] = If[n==0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, b[n, i - 1] + If[i>n, {0, 0, 0, 0}, Function[p, If[Mod[i, 2]==0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n-i, i]]]]]; a[n_] := b[2*n, 2*n][[3]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 27 2015, after Alois P. Heinz *)

More terms from and definition corrected by Alois P. Heinz, Feb 16 2014

A236914 Number of partitions of 2n+1 of type OO (see Comments).

Original entry on oeis.org

0, 1, 3, 7, 14, 27, 49, 86, 146, 242, 392, 623, 973, 1498, 2274, 3411, 5059, 7427, 10801, 15572, 22267, 31602, 44533, 62338, 86716, 119918, 164903, 225566, 306993, 415814, 560641, 752622, 1006132, 1339677, 1776980, 2348384, 3092594, 4058848, 5309608, 6923959

Offset: 0

Clark Kimberling, Feb 01 2014

The partitions of n are partitioned into four types:
EO, even # of odd parts and odd  # of even parts, A236559;
OE, odd  # of odd parts and even # of even parts, A160786;
EE, even # of odd parts and even # of even parts, A236913;
OO, odd  # of odd parts and odd  # of even parts, A236914.
A236559 and A160786 are the bisections of A027193;
A236913 and A236914 are the bisections of A027187.

			The partitions of 5 of type OO are [4,1], [3,2], [2,1,1,1], so that a(2) = 3.
type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11
EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0
OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29
EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0
OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000041, A000701, A027187, A027193, A160786, A236559, A236913.

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
      `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
      `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
          [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
    end:
a:= n-> b(2*n+1$2)[4]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 16 2014

z = 25; m1 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,  OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]]; m2 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,       OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]]; m3 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];
m1 (* A236559, type EO*)
m2 (* A160786, type OE*)
m3 (* A236913, type EE*)
m4 (* A236914, type OO*) (* Peter J. C. Moses, Feb 03 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[i > n, {0, 0, 0, 0}, Function[p, If[Mod[i, 2] == 0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n-i, i]]]]]; a[n_] := b[2*n+1, 2*n+1][[4]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 27 2015, after Alois P. Heinz *)

More terms from Alois P. Heinz, Feb 16 2014

A160786 The number of odd partitions of consecutive odd integers.

Original entry on oeis.org

1, 2, 4, 8, 16, 29, 52, 90, 151, 248, 400, 632, 985, 1512, 2291, 3431, 5084, 7456, 10836, 15613, 22316, 31659, 44601, 62416, 86809, 120025, 165028, 225710, 307161, 416006, 560864, 752877, 1006426, 1340012, 1777365, 2348821, 3093095, 4059416, 5310255, 6924691

Offset: 0

Utpal Sarkar (doetoe(AT)gmail.com), May 26 2009

nonn

It seems that these are partitions of odd length and sum, ranked by A340931. The parts do not have to be odd. - Gus Wiseman, Apr 06 2021

			From _Gus Wiseman_, Apr 06 2021: (Start)
The a(0) = 1 through a(4) = 16 partitions:
  (1)  (3)    (5)      (7)        (9)
       (111)  (221)    (322)      (333)
              (311)    (331)      (432)
              (11111)  (421)      (441)
                       (511)      (522)
                       (22111)    (531)
                       (31111)    (621)
                       (1111111)  (711)
                                  (22221)
                                  (32211)
                                  (33111)
                                  (42111)
                                  (51111)
                                  (2211111)
                                  (3111111)
                                  (111111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000
V. I. Arnold, On teaching mathematics

Partitions with all odd parts are counted by A000009 and ranked by A066208.
This is a bisection of A027193 (odd-length partitions), which is ranked by A026424.
The case of all odd parts is counted by A078408 and ranked by A300272.
The even version is A236913, ranked by A340784.
A multiplicative version is A340102.
These partitions are ranked by A340931.
A047993 counts balanced partitions, ranked by A106529.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A236914 counts partition of type OO, ranked by A341448.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Cf. A000700, A168659, A200750, A340604, A340607, A340854, A340855.

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
      `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
      `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
          [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
    end:
a:= n-> b(2*n+1$2)[2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 16 2014

b[n_, i_] := b[n, i] = If[n==0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, b[n, i-1] + If[i>n, {0, 0, 0, 0}, Function[{p}, If[Mod[i, 2]==0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n-i, i]]]]]; a[n_] := b[2*n+1, 2*n+1][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)
(* Slow but easy to read *)
a[n_] := Length@IntegerPartitions[2 n + 1, {1, 2 n + 1, 2}]
a /@ Range[0, 25]
(* Leo C. Stein, Nov 11 2020 *)
(* Faster, don't build the partitions themselves *)
(* Number of partitions of n into exactly k parts *)
P[0, 0] = 1;
P[n_, k_] := 0 /; ((k <= 0) || (n <= 0))
P[n_, k_] := P[n, k] = P[n - k, k] + P[n - 1, k - 1]
a[n_] := Sum[P[2 n + 1, k], {k, 1, 2 n + 1, 2}]
a /@ Range[0, 40]
(* Leo C. Stein, Nov 11 2020 *)

# Could be memoized for speedup
def numoddpart(n, m=1):
    """The number of partitions of n into an odd number of parts of size at least m"""
    if n < m:
        return 0
    elif n == m:
        return 1
    else:
        # 1 (namely n = n) and all partitions of the form
        # k + even partitions that start with >= k
        return 1 + sum([numevenpart(n - k,  k) for k in range(m, n//3 + 1)])
def numevenpart(n, m=1):
    """The number of partitions of n into an even number of parts of size at least m"""
    if n < 2*m:
        return 0
    elif n == 2*m:
        return 1
    else:
        return sum([numoddpart(n - k,  k) for k in range(m,  n//2 + 1)])
[numoddpart(n) for n in range(1, 70, 2)]

# dict to memoize
ps = {(0,0): 1}
def p(n, k):
    """Number of partitions of n into exactly k parts"""
    if (n,k) in ps: return ps[(n,k)]
    if (n<=0) or (k<=0): return 0
    ps[(n,k)] = p(n-k,k) + p(n-1,k-1)
    return ps[(n,k)]
def a(n): return sum([p(2*n+1, k) for k in range(1,2*n+3,2)])
[a(n) for n in range(0,41)]
# Leo C. Stein, Nov 11 2020

a(n) = A027193(2n+1).

A345913 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum >= 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82

Offset: 1

Gus Wiseman, Jul 04 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of terms together with the corresponding compositions begins:
     0: ()           17: (4,1)          37: (3,2,1)
     1: (1)          18: (3,2)          38: (3,1,2)
     2: (2)          19: (3,1,1)        39: (3,1,1,1)
     3: (1,1)        21: (2,2,1)        41: (2,3,1)
     4: (3)          22: (2,1,2)        42: (2,2,2)
     5: (2,1)        23: (2,1,1,1)      43: (2,2,1,1)
     7: (1,1,1)      26: (1,2,2)        44: (2,1,3)
     8: (4)          28: (1,1,3)        45: (2,1,2,1)
     9: (3,1)        29: (1,1,2,1)      46: (2,1,1,2)
    10: (2,2)        31: (1,1,1,1,1)    47: (2,1,1,1,1)
    11: (2,1,1)      32: (6)            50: (1,3,2)
    13: (1,2,1)      33: (5,1)          52: (1,2,3)
    14: (1,1,2)      34: (4,2)          53: (1,2,2,1)
    15: (1,1,1,1)    35: (4,1,1)        55: (1,2,1,1,1)
    16: (5)          36: (3,3)          56: (1,1,4)

These compositions are counted by A116406.
These are the positions of terms >= 0 in A124754.
The version for prime indices is A344609.
The reverse-alternating sum version is A345914.
The opposite (k <= 0) version is A345915.
The strict (k > 0) version is A345917.
The complement is A345919.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A008549, A025047, A027187, A032443, A034871, A114121, A163493, A236913, A238279, A344607, A344608, A344610, A344611.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]>=0&]

A347439 Number of factorizations of n with integer reciprocal alternating product.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 6, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 1, 4, 0, 0, 0, 1, 0, 0, 0, 5

Offset: 1

Gus Wiseman, Sep 07 2021

nonn

All of these factorizations have an even number of factors, so their reverse-alternating product is also an integer.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
The value of a(n) does not depend solely on the prime signature of n. See the example comparing a(144) and a(400). - Antti Karttunen, Jul 28 2024

			The a(n) factorizations for
n    = 16,       36,       64,           72,       128,          144:
a(n) = 3,        4,        6,            5,        7,            11
--------------------------------------------------------------------------------
       2*8       6*6       8*8           2*36      2*64          2*72
       4*4       2*18      2*32          3*24      4*32          3*48
       2*2*2*2   3*12      4*16          6*12      8*16          4*36
                 2*2*3*3   2*2*2*8       2*2*3*6   2*2*4*8       6*24
                           2*2*4*4       2*3*3*4   2*4*4*4       12*12
                           2*2*2*2*2*2             2*2*2*16      2*2*6*6
                                                   2*2*2*2*2*4   2*3*3*8
                                                                 3*3*4*4
                                                                 2*2*2*18
                                                                 2*2*3*12
                                                                 2*2*2*2*3*3
From _Antti Karttunen_, Jul 28 2024 (Start)
For n=400, there are 12 such factorizations:
  2*200
  4*100
  5*80
  10*40
  20*20
  2*2*2*50
  2*2*5*20
  2*2*10*10
  2*4*5*10
  2*5*5*8
  4*4*5*5
  2*2*2*2*5*5.
Note that 400 = 2^4 * 5^2 has the same prime signature as 144 = 2^4 * 3^2. 400 = 2*4*5*10 is the factorization for which there is no analogous factorization of 144, as 2*3*4*6 doesn't satisfy the condition of having an integer reciprocal alternating product.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Positions of 0's are A005117 \ {1}.
Positions of non-0's are 1 and A013929.
The restriction to powers of 2 is A027187, reverse A035363.
Positions of 1's are 1 and A082293.
The additive version is A119620, ranked by A347451 and A028982.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reciprocal version is A347437.
The reverse version is A347438.
Allowing any alternating product < 1 gives A347440.
The non-reciprocal reverse version is A347442.
Allowing any alternating product >= 1 gives A347456.
The restriction to perfect squares is A347459, non-reciprocal A347458.
A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A236913, A316523, A330972, A332269, A344606, A344607, A347445, A347446, A347454, A347457, A347463.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i,{i,Length[q]}];
Table[Length[Select[facs[n],IntegerQ[recaltprod[#]]&]],{n,100}]

A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

A347439(n, m=0, ap=1, e=1) = if(1==n, 1==denominator(ap), sumdiv(n, d, if(d>1 && d>=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(2^n) = A027187(n).

a(n^2) = A347459(n).

Data section extended up to a(108) by Antti Karttunen, Jul 28 2024

A347442 Number of factorizations of n with integer reverse-alternating product.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 8, 2, 3, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 8, 5, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 9, 1, 3, 3, 8, 1, 1, 1, 3, 1, 1, 1, 12

Offset: 1

Gus Wiseman, Sep 08 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(n) factorizations for n = 4, 8, 16, 32, 36, 54, 64:
  (4)    (8)      (16)       (32)         (36)       (54)     (64)
  (2*2)  (2*4)    (2*8)      (4*8)        (6*6)      (3*18)   (8*8)
         (2*2*2)  (4*4)      (2*16)       (2*18)     (2*3*9)  (2*32)
                  (2*2*4)    (2*2*8)      (3*12)     (3*3*6)  (4*16)
                  (2*2*2*2)  (2*4*4)      (2*2*9)             (2*4*8)
                             (2*2*2*4)    (2*3*6)             (4*4*4)
                             (2*2*2*2*2)  (3*3*4)             (2*2*16)
                                          (2*2*3*3)           (2*2*2*8)
                                                              (2*2*4*4)
                                                              (2*2*2*2*4)
                                                              (2*2*2*2*2*2)

Antti Karttunen, Table of n, a(n) for n = 1..65537

The restriction to powers of 2 is A000041, reverse A344607.
Positions of 2's are A001248.
Positions of 1's are A005117.
Positions of non-1's are A013929.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reverse version is A347437.
The reciprocal version is A347438.
The even-length case is A347439.
Allowing any alternating product < 1 gives A347440.
The odd-length case is A347441, ranked by A347453.
The additive version is A347445, ranked by A347457.
The non-reverse additive version is A347446, ranked by A347454.
Allowing any alternating product >= 1 gives A347456.
The ordered version is A347463.
A038548 counts possible reverse-alternating products of factorizations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A273013 counts ordered factorizations of n^2 with alternating product 1.
Cf. A000290, A025047, A330972, A347443, A347449, A347451, A347458, A347459, A347460, A347462.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],IntegerQ@*revaltprod]],{n,100}]

A347442(n, m=n, ap=1, e=0) = if(1==n, 1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347442(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A000041(n).

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

cp's OEIS Frontend

A027187 Number of partitions of n into an even number of parts.

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A035363 Number of partitions of n into even parts.

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A339846 Number of even-length factorizations of n into factors > 1.

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A067661 Number of partitions of n into distinct parts such that number of parts is even.

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A236559 Number of partitions of 2n of type EO (see Comments).

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A236914 Number of partitions of 2n+1 of type OO (see Comments).

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