A067659 -id:A067659 - cp's OEIS frontend

A340832 Number of factorizations of n into factors > 1 with odd least factor.

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

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)     (3*36)     (135)      (3*60)     (3*84)
  (5*9)    (9*12)     (3*45)     (5*36)     (7*36)
  (3*15)   (3*4*9)    (5*27)     (9*20)     (9*28)
  (3*3*5)  (3*6*6)    (9*15)     (5*6*6)    (3*3*28)
           (3*3*12)   (3*5*9)    (3*3*20)   (3*4*21)
           (3*3*3*4)  (3*3*15)   (3*4*15)   (3*6*14)
                      (3*3*3*5)  (3*5*12)   (3*7*12)
                                 (3*6*10)   (3*3*4*7)
                                 (3*3*4*5)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A340854.
Positions of nonzero terms are A340855.
The version for partitions is A026804.
Odd-length factorizations are counted by A339890.
The version looking at greatest factor is A340831.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340607 counts factorizations with odd length and greatest factor.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A026424 lists numbers with odd Omega.
A027193 counts partitions of odd length.
A058695 counts partitions of odd numbers (A300063).
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A244991 lists numbers whose greatest prime index is odd.
A340692 counts partitions of odd rank.
Cf. A050320, A160786, A340385, A340596, A340599, A340654, A340655, A340931.

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],OddQ@*Min]],{n,100}]

A340832(n, m=n, fc=1) = if(1==n, (m%2)&&!fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A340832(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

A332304 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is odd.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 7, 7, 13, 19, 25, 31, 43, 49, 61, 193, 205, 337, 475, 727, 985, 1363, 1741, 2359, 2983, 3841, 4705, 5929, 12193, 13777, 20527, 27631, 39901, 52651, 75601, 99151, 132907, 172297, 227053, 287569, 373525, 465241, 587563, 725839, 899761, 1457683

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(6) = 7 because we have [6], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A027193, A032020, A067659, A166444, A332305.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

nmax = 45; CoefficientList[Series[Sum[(2 k - 1)! x^(k (2 k - 1))/Product[1 - x^j, {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) = A032020(n) - A332305(n).

A359899 Number of strict odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 6, 1, 1, 6, 1, 5, 7, 1, 1, 8, 12, 1, 9, 2, 1, 33, 1, 1, 11, 1, 50, 12, 1, 1, 13, 70, 1, 46, 1, 1, 122, 1, 1, 16, 102, 155, 17, 1, 1, 30, 216, 258, 19, 1, 1, 310, 1, 1, 666, 1, 382, 23, 1, 1, 23, 1596, 1, 393, 1, 1

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(30) = 33 partitions:
  (30)  (11,10,9)  (8,7,6,5,4)
        (12,10,8)  (9,7,6,5,3)
        (13,10,7)  (9,8,6,4,3)
        (14,10,6)  (9,8,6,5,2)
        (15,10,5)  (10,7,6,4,3)
        (16,10,4)  (10,7,6,5,2)
        (17,10,3)  (10,8,6,4,2)
        (18,10,2)  (10,8,6,5,1)
        (19,10,1)  (10,9,6,3,2)
                   (10,9,6,4,1)
                   (11,7,6,4,2)
                   (11,7,6,5,1)
                   (11,8,6,3,2)
                   (11,8,6,4,1)
                   (11,9,6,3,1)
                   (12,7,6,3,2)
                   (12,7,6,4,1)
                   (12,8,6,3,1)
                   (12,9,6,2,1)
                   (13,7,6,3,1)
                   (13,8,6,2,1)
                   (14,7,6,2,1)
                   (11,10,6,2,1)

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

Strict odd-length case of A240219, complement A359894, ranked by A359889.
Strict case of A359895, complement A359896, ranked by A359891.
Odd-length case of A359897, complement A359898.
The complement is counted by A359900.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A240851, A359903, A359906, A359910.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]

\\ Q(n,k,m) is g.f. for k strict parts of max size m.
Q(n,k,m)={polcoef(prod(i=1, m, 1 + y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)); if(r>=h*(h+1), polcoef(Q(r, h, m-1)*Q(r, h, r), r)))))} \\ Andrew Howroyd, Jan 21 2023

a(p) = 1 for prime p. - Andrew Howroyd, Jan 21 2023

A361860 Number of integer partitions of n whose median part is the smallest.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 12, 15, 21, 25, 36, 44, 58, 72, 95, 117, 150, 185, 235, 289, 362, 441, 550, 670, 824, 1000, 1223, 1476, 1795, 2159, 2609, 3126, 3758, 4485, 5369, 6388, 7609, 9021, 10709, 12654, 14966, 17632, 20782, 24414, 28684, 33601, 39364, 45996

Offset: 1

Gus Wiseman, Apr 02 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For mean instead of median we have A000005.
For length instead of median we have A006141.
For maximum instead of median we have A053263.
For half-median we have A361861.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A053263, A067659, A111907, A116608, A118096, A237753, A240219, A359907, A361848, A361849.

Table[Length[Select[IntegerPartitions[n],Min@@#==Median[#]&]],{n,30}]

A340603 Heinz numbers of integer partitions of odd rank.

Original entry on oeis.org

3, 4, 7, 10, 12, 13, 15, 16, 18, 19, 22, 25, 27, 28, 29, 33, 34, 37, 40, 42, 43, 46, 48, 51, 52, 53, 55, 60, 61, 62, 63, 64, 69, 70, 71, 72, 76, 77, 78, 79, 82, 85, 88, 89, 90, 93, 94, 98, 100, 101, 105, 107, 108, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of partitions with their Heinz numbers begins:
      3: (2)           33: (5,2)           63: (4,2,2)
      4: (1,1)         34: (7,1)           64: (1,1,1,1,1,1)
      7: (4)           37: (12)            69: (9,2)
     10: (3,1)         40: (3,1,1,1)       70: (4,3,1)
     12: (2,1,1)       42: (4,2,1)         71: (20)
     13: (6)           43: (14)            72: (2,2,1,1,1)
     15: (3,2)         46: (9,1)           76: (8,1,1)
     16: (1,1,1,1)     48: (2,1,1,1,1)     77: (5,4)
     18: (2,2,1)       51: (7,2)           78: (6,2,1)
     19: (8)           52: (6,1,1)         79: (22)
     22: (5,1)         53: (16)            82: (13,1)
     25: (3,3)         55: (5,3)           85: (7,3)
     27: (2,2,2)       60: (3,2,1,1)       88: (5,1,1,1)
     28: (4,1,1)       61: (18)            89: (24)
     29: (10)          62: (11,1)          90: (3,2,2,1)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
These partitions are counted by A340692.
The complement is A340602, counted by A340601.
The case of positive rank is A340604.
- Rank -
A001222 gives number of prime indices.
A047993 counts partitions of rank 0 (A106529).
A061395 gives maximum prime index.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length (A026424).
A027193 (also) counts partitions of odd maximum (A244991).
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340102 counts odd-length factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340608, A340609, A340610.

Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&]

A061395(a(n)) - A001222(a(n)) is odd.

A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 3, 3, 2, 1, 0, 1, 0, 1, 4, 4, 3, 2, 1, 0, 1, 0, 1, 5, 6, 4, 3, 2, 1, 0, 1, 0, 1, 7, 7, 6, 4, 3, 2, 1, 0, 1, 0, 1, 8, 10, 8, 6, 4, 3, 2, 1, 0, 1

Offset: 0

Gus Wiseman, Jun 05 2021

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of 2n into an odd number of parts whose conjugate has exactly 2k odd parts.

Also the number of reversed strict integer partitions of 2n with alternating sum 2k.

			Triangle begins:
   1
   0   1
   0   0   1
   0   1   0   1
   0   1   1   0   1
   0   1   2   1   0   1
   0   1   3   2   1   0   1
   0   1   3   3   2   1   0   1
   0   1   4   4   3   2   1   0   1
   0   1   5   6   4   3   2   1   0   1
   0   1   7   7   6   4   3   2   1   0   1
   0   1   8  10   8   6   4   3   2   1   0   1
   0   1  10  13  12   8   6   4   3   2   1   0   1
   0   1  11  18  15  12   8   6   4   3   2   1   0   1
   0   1  14  22  21  16  12   8   6   4   3   2   1   0   1
   0   1  15  29  27  23  16  12   8   6   4   3   2   1   0   1
Row n = 8 counts the following partitions (empty columns indicated by dots):
  .  (8,7,1)  (7,6,3)      (7,5,4)   (9,4,3)   (11,3,2)  (13,2,1)  .  (16)
              (8,6,2)      (8,5,3)   (10,4,2)  (12,3,1)
              (9,6,1)      (9,5,2)   (11,4,1)
              (6,4,3,2,1)  (10,5,1)
Row n = 9 counts the following partitions (empty columns indicated by dots, A..I = 10..18):
  .  981   873     765     954   B43   D32   F21   .  I
           972     864     A53   C42   E31
           A71     963     B52   D41
           65421   A62     C51
           75321   B61
                   84321

The non-reversed version is A152146.
The non-reversed non-strict version is A239830.
Column k = 2 is A343941.
The non-strict version is A344610.
Row sums are A344650.
Right half of even-indexed rows of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sum of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A003242, A006330, A027187, A114121, A116406, A239829, A344607, A344608, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,30,2},{k,0,n,2}]

A359900 Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 4, 8, 10, 8, 15, 18, 17, 26, 27, 31, 43, 51, 53, 59, 81, 87, 109, 127, 115, 169, 194, 213, 255, 243, 322, 379, 431, 478, 487, 629, 667, 804, 907, 902, 1151, 1294, 1439, 1530, 1674, 2031, 2290, 2559, 2829, 2973, 3296, 3939

Offset: 0

Gus Wiseman, Jan 21 2023

nonn

			The a(7) = 1 through a(16) = 15 partitions (A=10, B=11, C=12, D=13):
  (421)  (431)  (621)  (532)  (542)  (651)  (643)  (653)  (762)  (754)
         (521)         (541)  (632)  (732)  (652)  (743)  (843)  (763)
                       (631)  (641)  (831)  (742)  (752)  (861)  (853)
                       (721)  (731)  (921)  (751)  (761)  (942)  (862)
                              (821)         (832)  (842)  (A32)  (871)
                                            (841)  (851)  (A41)  (943)
                                            (931)  (932)  (B31)  (952)
                                            (A21)  (941)  (C21)  (961)
                                                   (A31)         (A42)
                                                   (B21)         (A51)
                                                                 (B32)
                                                                 (B41)
                                                                 (C31)
                                                                 (D21)
                                                                 (64321)

This is the strict case of A359896, complement A359895, ranked by A359892.
This is the odd-length case of A359898, complement A359897.
The complement is counted by A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A359893/A359901/A359902 count partitions by median, ranked by A360005.
Cf. A000016, A065795, A066571, A102627, A240850, A240851, A327475, A359894, A359906, A359907, A359910.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]!=Median[#]&]],{n,0,30}]

A347449 Number of integer partitions of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 10, 11, 20, 22, 37, 41, 66, 75, 113, 129, 190, 218, 310, 358, 497, 576, 782, 908, 1212, 1411, 1851, 2156, 2793, 3255, 4163, 4853, 6142, 7159, 8972, 10451, 12989, 15123, 18646, 21689, 26561, 30867, 37556, 43599, 52743, 61161, 73593

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

All such partitions have odd length.

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(2) = 1 through a(9) = 11 partitions:
  (2)  (3)  (4)    (5)    (6)      (7)      (8)        (9)
            (211)  (311)  (222)    (322)    (332)      (333)
                          (321)    (421)    (422)      (432)
                          (411)    (511)    (431)      (522)
                          (21111)  (31111)  (521)      (531)
                                            (611)      (621)
                                            (22211)    (711)
                                            (32111)    (32211)
                                            (41111)    (42111)
                                            (2111111)  (51111)
                                                       (3111111)

The strict case is A067659, except that a(0) = a(1) = 0.
The even bisection is A236559.
The non-reverse multiplicative version is A339890, weak A347456.
The case of >= 1 instead of > 1 is A344607.
The opposite version is A344608, also the non-reverse even-length case.
The complement is counted by A347443, non-reverse A119620.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
Reverse version of A347448; also the odd-length case.
The Heinz numbers of these partitions are the complement of A347450.
The multiplicative version (factorizations) is A347705.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A100824 counts partitions of n with alternating sum <= 1.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347462 counts possible reverse-alternating products of partitions.
Cf. A000070, A008549, A086543, A182616, A236913, A325534, A325535, A344611, A347442, A347444, A347447, A347453, A347461, A347465.

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

a(n) = A344607(n) - A119620(n).

A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

Original entry on oeis.org

0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004

Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos, Mar 04 2004

Occurs also as first column of a matrix-inversion occurring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007

			From _Gus Wiseman_, Jan 06 2021: (Start)
a(n) is the number of ordered set partitions of {1..n} into an odd number of blocks. The a(1) = 1 through a(3) = 7 ordered set partitions are:
  {{1}}  {{1,2}}  {{1,2,3}}
                  {{1},{2},{3}}
                  {{1},{3},{2}}
                  {{2},{1},{3}}
                  {{2},{3},{1}}
                  {{3},{1},{2}}
                  {{3},{2},{1}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J.-C. Aval, V. Féray, J.-C. Novelli, J.-Y. Thibon, Quasi-symmetric functions as polynomial functions on Young diagrams, arXiv preprint arXiv:1312.2727, 2013
Gottfried Helms, Discussion of a problem concerning summing of like powers

Ordered set partitions are counted by A000670.
The case of (unordered) set partitions is A024429.
The complement (even-length ordered set partitions) is counted by A052841.
A058695 counts partitions of odd numbers, ranked by A300063.
A101707 counts partitions of odd positive rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340102 counts odd-length factorizations into odd factors.
A340692 counts partitions of odd rank.
Other cases of odd length:
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
Cf. A000700, A026424, A027187, A028260, A078408, A174725, A236914, A340101.

h := n -> add(combinat:-eulerian1(n,k)*2^k,k=0..n):
a := n -> (h(n)-(-1)^n)/2: seq(a(n),n=0..20); # Peter Luschny, Jul 09 2015

Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]

a(n)=if(n<0,0,n!*polcoeff(subst(y/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{a(n)=polcoeff(sum(m=0,n,(2*m+1)!*x^(2*m+1)/prod(k=1,2*m+1,1-k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

def A089677_list(len):  # with a(0)=1
    e, r = [1], [1]
    for i in (1..len-1):
        for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)
        r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))
        e.append(sum(e))
    return r
A089677_list(21) # Peter Luschny, Jul 09 2015

E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))).
O.g.f.: Sum_{n>=0} (2*n+1)! * x^(2*n+1) / Product_{k=1..2*n+1} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic, Jan 17 2005
a(n) ~ n! / (4*(log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015

A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)      (4*27)        (135)       (4*45)        (4*63)
  (5*9)     (2*6*9)       (3*45)      (12*15)       (12*21)
  (3*15)    (3*4*9)       (5*27)      (4*5*9)       (4*7*9)
  (3*3*5)   (2*2*27)      (9*15)      (2*2*45)      (6*6*7)
            (2*2*3*9)     (3*5*9)     (2*6*15)      (2*2*63)
            (2*2*3*3*3)   (3*3*15)    (3*4*15)      (2*6*21)
                          (3*3*3*5)   (2*2*5*9)     (3*4*21)
                                      (3*3*4*5)     (2*2*7*9)
                                      (2*2*3*15)    (2*3*6*7)
                                      (2*2*3*3*5)   (3*3*4*7)
                                                    (2*2*3*21)
                                                    (2*2*3*3*7)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A000079.
The version for partitions is A027193.
The version for prime indices is A244991.
The version looking at length instead of greatest factor is A339890.
The version that also has odd length is A340607.
The version looking at least factor is A340832.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A058695 counts partitions of odd numbers.
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A050320, A061395, A339846, A340385, A340596, A340599, A340654, A340655, A340785, A340854, A340855.

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],OddQ@*Max]],{n,100}]

A340831(n, m=n, fc=1) = if(1==n, !fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(!fc||(d%2)), s += A340831(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

cp's OEIS Frontend

A340832 Number of factorizations of n into factors > 1 with odd least factor.

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A332304 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is odd.

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A359899 Number of strict odd-length integer partitions of n whose parts have the same mean as median.

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A361860 Number of integer partitions of n whose median part is the smallest.

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A340603 Heinz numbers of integer partitions of odd rank.

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A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.

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A359900 Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.

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A347449 Number of integer partitions of n with reverse-alternating product > 1.

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