A067661 -id:A067661 - cp's OEIS frontend

A238450 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 1, 0, 0, 1, 3, 2, 2, 1, 2, 1, 1, 0, 0, 1, 3, 3, 2, 2, 1, 2, 1, 1, 0, 0, 1, 4, 3, 3, 3, 2, 2, 2, 1, 1, 0, 0, 1, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1

Offset: 1

Mircea Merca, Feb 26 2014

			n\k | 1 2 3 4 5 6 7 8 9 10
   1: 1
   2: 0 1
   3: 0 0 1
   4: 0 0 0 1
   5: 0 0 0 0 1
   6: 1 1 1 0 0 1
   7: 1 1 0 1 0 0 1
   8: 2 1 1 1 1 0 0 1
   9: 2 2 2 1 1 1 0 0 1
  10: 3 2 2 1 2 1 1 0 0 1

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=1..6 are A238208, A238209, A238210, A238211, A238212, A238213.
Row sums are A238131.
Cf. A015716, A067659, A067661, A238451.

T(n,k) = {my(m=n-k); if(m>0, polcoef(prod(j=1, m, 1+x^j + O(x*x^m))/(1+x^k) + prod(j=1, m, 1-x^j + O(x*x^m))/(1-x^k), m)/2, m==0)} \\ Andrew Howroyd, Apr 29 2020

T(n,k) = Sum_{j=1..round(n/(2*k))} A067661(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067659(n-2*j*k).
G.f. of column k: (1/2)*(q^k/(1+q^k))*(-q;q){inf} + (1/2)*(q^k/(1-q^k))*(q;q){inf}.
T(n,k) = A015716(n,k) - A238451(n,k). - Andrew Howroyd, Apr 29 2020

Terms a(79) and beyond from Andrew Howroyd, Apr 29 2020

A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 0, 1, 1, 1, 1, 0, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 0, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 0

Offset: 1

Mircea Merca, Feb 26 2014

			n/k | 1 2 3 4 5 6 7 8 9 10
   1: 0
   2: 0 0
   3: 1 1 0
   4: 1 0 1 0
   5: 1 1 1 1 0
   6: 1 1 0 1 1 0
   7: 1 1 1 1 1 1 0
   8: 1 1 1 0 1 1 1 0
   9: 1 1 1 1 1 1 1 1 0
  10: 2 2 2 2 0 1 1 1 1 0

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=1..6 are A238215, A238217, A238218, A238219, A238220, A238221.
Row sums are A238132.
Cf. A015716, A067659, A067661, A238450.

T(n,k) = {my(m=n-k); if(m>0, polcoef(prod(j=1, m, 1+x^j + O(x*x^m))/(1+x^k) - prod(j=1, m, 1-x^j + O(x*x^m))/(1-x^k), m)/2, 0)} \\ Andrew Howroyd, Apr 29 2020

T(n,k) = Sum_{j=1..round(n/(2*k))} A067659(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067661(n-2*j*k).
G.f. of column k: (1/2)*(q^k/(1+q^k))*(-q;q){inf} - (1/2)*(q^k/(1-q^k))*(q;q){inf}.
T(n,k) = A015716(n,k) - A238450(n,k). - Andrew Howroyd, Apr 29 2020

A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

Original entry on oeis.org

3, 7, 9, 13, 15, 19, 21, 27, 29, 33, 37, 39, 43, 45, 49, 51, 53, 57, 61, 63, 69, 71, 75, 77, 79, 81, 87, 89, 91, 93, 99, 101, 105, 107, 111, 113, 117, 119, 123, 129, 131, 133, 135, 139, 141, 147, 151, 153, 159, 161, 163, 165, 169, 171, 173, 177, 181, 183

Offset: 1

Gus Wiseman, Feb 12 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. 1 has no prime indices so is not counted.

			The sequence of terms together with their prime indices begins:
      3: {2}         51: {2,7}         99: {2,2,5}
      7: {4}         53: {16}         101: {26}
      9: {2,2}       57: {2,8}        105: {2,3,4}
     13: {6}         61: {18}         107: {28}
     15: {2,3}       63: {2,2,4}      111: {2,12}
     19: {8}         69: {2,9}        113: {30}
     21: {2,4}       71: {20}         117: {2,2,6}
     27: {2,2,2}     75: {2,3,3}      119: {4,7}
     29: {10}        77: {4,5}        123: {2,13}
     33: {2,5}       79: {22}         129: {2,14}
     37: {12}        81: {2,2,2,2}    131: {32}
     39: {2,6}       87: {2,10}       133: {4,8}
     43: {14}        89: {24}         135: {2,2,2,3}
     45: {2,2,3}     91: {4,6}        139: {34}
     49: {4,4}       93: {2,11}       141: {2,15}

These partitions are counted by A026805.
Looking at length or at maximum gives A028260/A244990, counted by A027187.
If all prime indices are even we get A066207, counted by A035363.
The complement is {1} \/ A340932, counted by A026804.
A001222 counts prime factors.
A005843 lists even numbers.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058695 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A112798 lists the prime indices of each positive integer.
Cf. A006141, A030229, A067661, A101708, A236913,  A339846, A340601, A340602, A340605, A340854/A340855.

Select[Range[2,100],EvenQ[PrimePi[FactorInteger[#][[1,1]]]]&]

A055396(a(n)) belongs to A005843.

Closed under multiplication.

A340786 Number of factorizations of 4n into an even number of even factors > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 1, 7, 2, 2, 2, 7, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 3, 12, 2, 4, 1, 4, 2, 4, 1, 10, 1, 2, 3, 4, 2, 4, 1, 10, 3, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Jan 31 2021

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
  4*6   6*8      2*48      2*72      4*60      4*80          40*50
  2*12  2*24     4*24      4*36      6*40      8*40          4*500
        4*12     6*16      6*24      8*30      10*32         8*250
        2*2*2*6  8*12      8*18      10*24     16*20         10*200
                 2*2*4*6   12*12     12*20     2*160         20*100
                 2*2*2*12  2*2*6*6   2*120     2*2*2*40      2*1000
                           2*2*2*18  2*2*2*30  2*2*4*20      2*2*10*50
                                     2*2*6*10  2*2*8*10      2*2*2*250
                                               2*4*4*10      2*10*10*10
                                               2*2*2*2*2*10

Robert Israel, Table of n, a(n) for n = 1..10000

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of ones are 1 and A000040, or A008578.
A version for partitions is A027187 (A028260).
Allowing odd length gives A108501 (bisection of A340785).
Allowing odd factors gives A339846.
An odd version is A340102.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
- Even -
A027187 counts partitions of even maximum (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).
Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

g:= proc(n, m, p)
 option remember;
 local F,r,x,i;
 # number of factorizations of n into even factors > m with number of factors == p (mod 2)
 if n = 1 then if p = 0 then return 1 else return 0 fi fi;
 if m > n  or n::odd then return 0 fi;
 F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)),list));
 r:= 0;
 for x in F do
   for i from 1 while n mod x^i = 0 do
     r:= r + procname(n/x^i, x, (p+i) mod 2)
 od od;
 r
end proc:
f:= n -> g(4*n, 1, 0):
map(f, [$1..100]); # Robert Israel, Mar 16 2023

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[4n],EvenQ[Length[#]]&&Select[#,OddQ]=={}&]],{n,100}]

A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
A340786(n) = A340786aux(4*n); \\ Antti Karttunen, Apr 14 2022

A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k) / Product_{j=1..2k} (1 - x^j).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 15, 22, 34, 48, 70, 97, 137, 186, 255, 341, 459, 605, 800, 1042, 1359, 1751, 2256, 2879, 3672, 4645, 5869, 7367, 9234, 11508, 14319, 17730, 21916, 26975, 33143, 40570, 49575, 60376, 73402, 88974, 107666, 129933, 156546, 188148, 225767, 270300, 323115, 385453

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

Partial sums of A027187.
From Gus Wiseman, Jun 26 2021: (Start)
Also the number of integer partitions of 2n+1 with odd greatest part and alternating sum 1, where the alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition. For example, the a(0) = 1 through a(6) = 15 partitions are:
  1   111   32      331       54          551           76
            11111   3211      3222        3332          5422
                    1111111   3321        5411          5521
                              33111       33221         33331
                              321111      322211        55111
                              111111111   332111        322222
                                          3311111       332221
                                          32111111      333211
                                          11111111111   541111
                                                        3322111
                                                        32221111
                                                        33211111
                                                        331111111
                                                        3211111111
                                                        1111111111111
Also odd-length partitions of 2n+1 with exactly one odd part.
(End)

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.1 "Unrestricted partitions and partitions into m parts", page 347.
Index entries for sequences related to partitions

First differences are A027187.
The version for even instead of odd greatest part is A306145.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A067661 counts strict partitions of even length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000097, A006330, A027193, A030229, A067659, A236559, A236914, A239829, A239830, A318156, A338907, A344611.

nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k)/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 + EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 26 2021 *)

a(n) = A000070(n) - A306145(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Aug 20 2018

A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k+1) / Product_{j=1..2k+1} (1 - x^j).

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 15, 23, 33, 49, 69, 98, 135, 187, 253, 343, 456, 607, 797, 1045, 1355, 1755, 2252, 2884, 3666, 4651, 5863, 7375, 9226, 11517, 14310, 17741, 21904, 26988, 33130, 40586, 49558, 60394, 73383, 88996, 107642, 129958, 156519, 188178, 225734, 270335, 323078, 385494

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

Partial sums of A027193.
From Gus Wiseman, Jun 23 2021: (Start)
Also the number of even-length integer partitions of 2n+1 with exactly one odd part. For example, the a(1) = 1 through a(5) = 10 partitions are:
  (2,1)  (3,2)  (4,3)      (5,4)      (6,5)
         (4,1)  (5,2)      (6,3)      (7,4)
                (6,1)      (7,2)      (8,3)
                (2,2,2,1)  (8,1)      (9,2)
                           (3,2,2,2)  (10,1)
                           (4,2,2,1)  (4,3,2,2)
                                      (4,4,2,1)
                                      (5,2,2,2)
                                      (6,2,2,1)
                                      (2,2,2,2,2,1)
Also partitions of 2n+1 with even greatest part and alternating sum 1.
(End)

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.1 "Unrestricted partitions and partitions into m parts", page 347.
Index entries for sequences related to partitions

First differences are A027193.
The ordered version appears to be A087447 modulo initial terms.
The version for odd instead of even-length partitions is A304620.
The case of strict partitions is A318156.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions of even length, with strict case A067661.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A006330, A030229, A067659, A236559, A236914, A239829, A239830, A338907, A344611.

nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k + 1)/Product[(1 - x^j), {j, 1, 2 k + 1}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 23 2021 *)

a(n) = A000070(n) - A304620(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Aug 20 2018

A347704 Number of even-length integer partitions of n with integer alternating product.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 6, 4, 11, 8, 18, 13, 33, 22, 49, 38, 79, 58, 122, 90, 186, 139, 268, 206, 402, 304, 569, 448, 817, 636, 1152, 907, 1612, 1283, 2220, 1791, 3071, 2468, 4162, 3409, 5655, 4634, 7597, 6283, 10171, 8478, 13491, 11336, 17906, 15088, 23513, 20012

Offset: 0

Gus Wiseman, Sep 17 2021

nonn

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

			The a(2) = 1 through a(9) = 8 partitions:
  (11)  (21)  (22)    (41)    (33)      (61)      (44)        (63)
              (31)    (2111)  (42)      (2221)    (62)        (81)
              (1111)          (51)      (4111)    (71)        (3321)
                              (2211)    (211111)  (2222)      (4221)
                              (3111)              (3221)      (6111)
                              (111111)            (3311)      (222111)
                                                  (4211)      (411111)
                                                  (5111)      (21111111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)

Allowing any alternating product >= 1 gives A000041, reverse A344607.
Allowing any alternating product gives A027187, odd bisection A236914.
The Heinz numbers of these partitions are given by A028260 /\ A347457.
The reverse and reciprocal versions are both A035363.
The multiplicative version (factorizations) is A347438, reverse A347439.
The odd-length instead of even-length version is A347444.
Allowing any length gives A347446.
A034008 counts even-length compositions, ranked by A053754.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A067661, A236913, A304620, A339846, A347437, A347441, A347442, A347445, A347448, A347449, A347454, A347462.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

A339366 Number of partitions of n into an even number of distinct squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 3, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 3

Offset: 0

Ilya Gutkovskiy, Dec 01 2020

nonn

			a(50) = 2 because we have [49, 1] and [36, 9, 4, 1].

Cf. A000290, A033461, A067661, A276516, A339364, A339365, A339367.

nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] + Product[(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

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

a(n) = (A033461(n) + A276516(n)) / 2.

A343942 Number of even-length strict integer partitions of 2n+1.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 52, 71, 96, 128, 170, 224, 292, 380, 491, 630, 805, 1024, 1295, 1632, 2049, 2560, 3189, 3959, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29249, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937, 172928

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

By conjugation, also the number of integer partitions of 2n+1 covering an initial interval of positive integers with greatest part even.

			The a(1) = 1 through a(7) = 13 strict partitions:
  (2,1)  (3,2)  (4,3)  (5,4)  (6,5)      (7,6)      (8,7)
         (4,1)  (5,2)  (6,3)  (7,4)      (8,5)      (9,6)
                (6,1)  (7,2)  (8,3)      (9,4)      (10,5)
                       (8,1)  (9,2)      (10,3)     (11,4)
                              (10,1)     (11,2)     (12,3)
                              (5,3,2,1)  (12,1)     (13,2)
                                         (5,4,3,1)  (14,1)
                                         (6,4,2,1)  (6,4,3,2)
                                         (7,3,2,1)  (6,5,3,1)
                                                    (7,4,3,1)
                                                    (7,5,2,1)
                                                    (8,4,2,1)
                                                    (9,3,2,1)

Ranked by A005117 (strict), A028260 (even length), and A300063 (odd sum).
Odd bisection of A067661 (non-strict: A027187).
The non-strict version is A236914.
The opposite type of strict partition (odd length and even sum) is A344650.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A030229, A035294, A067659, A236559, A338907, A343941, A344649, A344654, A344739.

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,15}]

The Heinz numbers are A005117 /\ A028260 /\ A300063.

More terms from Bert Dobbelaere, Jun 12 2021

A339375 Number of partitions of n into an even number of distinct triangular numbers.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(31) = 3 because we have [28, 3], [21, 10] and [21, 6, 3, 1].

Cf. A000217, A024940, A067661, A292518, A339366, A339373, A339374, A339376.

nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k (k + 1)/2)), {k, 1, nmax}] + Product[(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(n) = (A024940(n) + A292518(n)) / 2.

cp's OEIS Frontend

A238450 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

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A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

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A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

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Mathematica

Formula

A340786 Number of factorizations of 4n into an even number of even factors > 1.

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Maple

Mathematica

PARI

A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k) / Product_{j=1..2*k} (1 - x^j).

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Mathematica

Formula

A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k+1) / Product_{j=1..2*k+1} (1 - x^j).

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Mathematica

Formula

A347704 Number of even-length integer partitions of n with integer alternating product.

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Mathematica

A339366 Number of partitions of n into an even number of distinct squares.

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A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k) / Product_{j=1..2k} (1 - x^j).

A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k+1) / Product_{j=1..2k+1} (1 - x^j).