A078408 -id:A078408 - cp's OEIS frontend

A282892 The difference between the number of partitions of n into odd parts (A000009) and the number of partitions of n into even parts (A035363).

0, 1, 0, 2, 0, 3, 1, 5, 1, 8, 3, 12, 4, 18, 7, 27, 10, 38, 16, 54, 22, 76, 33, 104, 45, 142, 64, 192, 87, 256, 120, 340, 159, 448, 215, 585, 283, 760, 374, 982, 486, 1260, 634, 1610, 814, 2048, 1049, 2590, 1335, 3264, 1700, 4097, 2146, 5120, 2708, 6378, 3390, 7917, 4243, 9792, 5276

Offset: 0

Robert G. Wilson v, Feb 24 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..165 from Robert G. Wilson v)

Cf. A000009, A035363, A078408, A282893.

with(numtheory):
b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
      (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
    end:
a:= n-> b(n, 0) -b(n, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 24 2017

f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[f, 60]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[Sum[If[
     OddQ[d+t], d, 0], {d, Divisors[j]}]*b[n-j, t], {j, 1, n}]/n];
a[n_] := b[n, 0] - b[n, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

a(2n-1) = A000009(2n-1) = A078408(n).

a(2n) = A282893(n).

A300351 Triangle whose n-th row lists in order all Heinz numbers of integer partitions of n into odd parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 11, 20, 32, 22, 25, 40, 64, 17, 44, 50, 80, 128, 34, 55, 88, 100, 160, 256, 23, 68, 110, 125, 176, 200, 320, 512, 46, 85, 121, 136, 220, 250, 352, 400, 640, 1024, 31, 92, 170, 242, 272, 275, 440, 500, 704, 800, 1280, 2048, 62, 115, 184

Offset: 1

Gus Wiseman, Mar 03 2018

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Triangle of partitions into odd parts begins:
                   0
                  (1)
                  (11)
                (3) (111)
               (31) (1111)
            (5) (311) (11111)
        (51) (33) (3111) (111111)
    (7) (511) (331) (31111) (1111111)
(71) (53) (5111) (3311) (311111) (11111111)

Cf. A000009, A031368, A056239, A066208, A078408, A215366, A246867, A299759, A299757, A300063, A300272.

Table[Sort[Times@@Prime/@#&/@Select[IntegerPartitions[n],And@@OddQ/@#&]],{n,0,12}]

A346634 Number of strict odd-length integer partitions of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937

Offset: 0

Gus Wiseman, Aug 01 2021

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
                          (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
                                   (7,3,1)  (7,5,1)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (9,3,1)   (9,4,2)
                                            (10,2,1)  (9,5,1)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (11,3,1)
                                                      (12,2,1)
                                                      (5,4,3,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Odd bisection of A067659, which is ranked by A030059.
The even version is the even bisection of A067661.
The case of all odd parts is counted by A069911 (non-strict: A078408).
The non-strict version is A160786, ranked by A340931.
The non-strict even version is A236913, ranked by A340784.
The even-length version is A343942 (non-strict: A236914).
The even-sum version is A344650 (non-strict: A236559 or A344611).
A000009 counts partitions with all odd parts, ranked by A066208.
A000009 counts strict partitions, ranked by A005117.
A027193 counts odd-length partitions, ranked by A026424.
A027193 counts odd-maximum partitions, ranked by A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Other cases of odd length:
- A024429 set partitions
- A089677 ordered set partitions
- A166444 compositions
- A174726 ordered factorizations
- A332304 strict compositions
- A339890 factorizations
Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

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(2*n+1$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

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

More terms from Alois P. Heinz, Aug 05 2021

A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 4, 8, 15, 25, 42, 68, 107, 166, 253, 377, 557, 811, 1166, 1661, 2344, 3275, 4543, 6253, 8544, 11600, 15653, 20994, 28011, 37178, 49100, 64550, 84489, 110115, 142951, 184867, 238196, 305844, 391391, 499244, 634865, 804925, 1017610, 1282957, 1613195

Offset: 0

Michael Somos, Oct 06 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 25*x^5 + 42*x^6 + 68*x^7 + ...
G.f. = q^13 + 2*q^29 + 4*q^45 + 8*q^61 + 15*q^77 + 25*q^93 + 42*q^109 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A078408, A097451, A132217, A262160.

a[ n_] := SeriesCoefficient[ - x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n + 1}];

{a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( - eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

Expansion of - (psi(x^6) / psi(x) - psi(x^6) / psi(-x)) / (2 * x) in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [ 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 1, 0, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 2, 0, ...].
a(n) = A132217(2*n + 1) = - A262160(2*n + 1).
Convolution product of A097451 and A078408.
a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Mar 31 2018

A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k(2k+1)) / Product_{j=1..2*k} (1 - x^j).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 35, 44, 55, 68, 84, 103, 126, 153, 185, 223, 268, 320, 381, 452, 535, 631, 742, 870, 1018, 1188, 1383, 1607, 1863, 2155, 2489, 2869, 3301, 3792, 4348, 4978, 5691, 6496, 7404, 8428, 9580, 10875, 12330, 13962, 15791, 17840, 20131, 22691

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

Partial sums of A067661.
From Gus Wiseman, Jul 29 2021: (Start)
Also the number of strict integer partitions of 2n+1 of odd length with exactly one odd part. For example, the a(1) = 1 through a(7) = 10 partitions are:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (6,2,1)  (6,3,2)  (6,5,2)   (7,6,2)
                                   (6,4,1)  (7,4,2)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (10,2,1)  (9,4,2)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (12,2,1)
The following relate to these partitions:
- Not requiring odd length gives A036469.
- The non-strict version is A304620.
- The version for even instead of odd length is A318156.
- Allowing any number of odd parts gives A346634 (bisection of A067659).
(End)

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

Cf. A036469, A078616, A304620, A318156.
First differences are A067661 (non-strict: A027187, odd bisection: A343942).
A000041 counts partitions.
A000070 counts partitions with alternating sum 1.
A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A027193, A035294, A067659, A087447, A236559, A236914, A239829, A306145, A344611, A344739, A346634.

nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k + 1))/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] + QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman, Jul 29 2021 *)

a(n) = A036469(n) - A318156(n).
a(n) = A318156(n) + A078616(n).
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 20 2018

A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 107, 336, 1037, 3367, 10924, 36438, 121045, 412789, 1398168, 4831708, 16636297, 58084208, 202101971, 712709423, 2502000811, 8880033929, 31428410158, 112199775788, 399383181020, 1433385148187, 5128572792587, 18481258241133

Offset: 0

Gus Wiseman, Aug 27 2018

nonn

A p-tree of weight n with odd outdegrees is either a single node (if n = 1) or a finite odd-length sequence of at least 3 p-trees with odd outdegrees whose weights are weakly decreasing and sum to n.

			The a(4) = 13 p-trees of weight 9 with odd outdegrees:
  ((((ooo)oo)oo)oo)
  (((ooo)(ooo)o)oo)
  (((ooo)oo)(ooo)o)
  ((ooo)(ooo)(ooo))
  (((ooooo)oo)oo)
  (((ooo)oooo)oo)
  ((ooooo)(ooo)o)
  (((ooo)oo)oooo)
  ((ooo)(ooo)ooo)
  ((ooooooo)oo)
  ((ooooo)oooo)
  ((ooo)oooooo)
  (ooooooooo)

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A027193, A063834, A078408, A196545, A279374, A289501, A298118, A300300, A300301, A300355, A300436, A300647, A300652, A300797, A302243.

b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Table[b[n],{n,1,20,2}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))) - 1/prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 27 2018

A341448 Heinz numbers of integer partitions of type OO.

Original entry on oeis.org

6, 14, 15, 24, 26, 33, 35, 38, 51, 54, 56, 58, 60, 65, 69, 74, 77, 86, 93, 95, 96, 104, 106, 119, 122, 123, 126, 132, 135, 140, 141, 142, 143, 145, 150, 152, 158, 161, 177, 178, 185, 201, 202, 204, 209, 214, 215, 216, 217, 219, 221, 224, 226, 232, 234, 240

Offset: 1

Gus Wiseman, Feb 15 2021

nonn

These partitions are defined to have an odd number of odd parts and an odd number of even parts. They also have even length and odd sum.

			The sequence of partitions together with their Heinz numbers begins:
      6: (2,1)         74: (12,1)           141: (15,2)
     14: (4,1)         77: (5,4)            142: (20,1)
     15: (3,2)         86: (14,1)           143: (6,5)
     24: (2,1,1,1)     93: (11,2)           145: (10,3)
     26: (6,1)         95: (8,3)            150: (3,3,2,1)
     33: (5,2)         96: (2,1,1,1,1,1)    152: (8,1,1,1)
     35: (4,3)        104: (6,1,1,1)        158: (22,1)
     38: (8,1)        106: (16,1)           161: (9,4)
     51: (7,2)        119: (7,4)            177: (17,2)
     54: (2,2,2,1)    122: (18,1)           178: (24,1)
     56: (4,1,1,1)    123: (13,2)           185: (12,3)
     58: (10,1)       126: (4,2,2,1)        201: (19,2)
     60: (3,2,1,1)    132: (5,2,1,1)        202: (26,1)
     65: (6,3)        135: (3,2,2,2)        204: (7,2,1,1)
     69: (9,2)        140: (4,3,1,1)        209: (8,5)

Note: A-numbers of ranking sequences are in parentheses below.
The case of odd parts, length, and sum is counted by A078408 (A300272).
The type EE version is A236913 (A340784).
These partitions (for odd n) are counted by A236914.
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd (A340932).
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A160786 counts odd-length partitions of odd numbers (A340931).
A340101 counts factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A000700, A024429, A027187, A106529, A117409, A174725, A257541, A325134, A339890, A340102, A340604.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[Count[primeMS[#],?EvenQ]]&&OddQ[Count[primeMS[#],?OddQ]]&]

A341449 Heinz numbers of integer partitions into odd parts > 1.

Original entry on oeis.org

1, 5, 11, 17, 23, 25, 31, 41, 47, 55, 59, 67, 73, 83, 85, 97, 103, 109, 115, 121, 125, 127, 137, 149, 155, 157, 167, 179, 187, 191, 197, 205, 211, 227, 233, 235, 241, 253, 257, 269, 275, 277, 283, 289, 295, 307, 313, 331, 335, 341, 347, 353, 365, 367, 379, 389

Offset: 1

Gus Wiseman, Feb 15 2021

nonn

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 together with their Heinz numbers begins:
      1: ()        97: (25)       197: (45)       307: (63)
      5: (3)      103: (27)       205: (13,3)     313: (65)
     11: (5)      109: (29)       211: (47)       331: (67)
     17: (7)      115: (9,3)      227: (49)       335: (19,3)
     23: (9)      121: (5,5)      233: (51)       341: (11,5)
     25: (3,3)    125: (3,3,3)    235: (15,3)     347: (69)
     31: (11)     127: (31)       241: (53)       353: (71)
     41: (13)     137: (33)       253: (9,5)      365: (21,3)
     47: (15)     149: (35)       257: (55)       367: (73)
     55: (5,3)    155: (11,3)     269: (57)       379: (75)
     59: (17)     157: (37)       275: (5,3,3)    389: (77)
     67: (19)     167: (39)       277: (59)       391: (9,7)
     73: (21)     179: (41)       283: (61)       401: (79)
     83: (23)     187: (7,5)      289: (7,7)      415: (23,3)
     85: (7,3)    191: (43)       295: (17,3)     419: (81)

Note: A-numbers of ranking sequences are in parentheses below.
Partitions with no ones are A002865 (A005408).
The case of even parts is A035363 (A066207).
These partitions are counted by A087897.
The version for factorizations is A340101.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A056239 adds up prime indices.
A078408 counts partitions with odd parts, length, and sum (A300272).
A112798 lists the prime indices of each positive integer.
A257991/A257992 count odd/even prime indices.
Cf. A026804 (A340932), A160786 (A340931), A340386, A340604, A340607, A340785, A340854/A340855, A341446.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&OddQ[Times@@primeMS[#]]&]

A365829 Squarefree non-semiprimes.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 181, 182, 186

Offset: 1

Gus Wiseman, Oct 05 2023

nonn

First differs from A030059 in having 210.

			The terms together with their prime indices begin:
     1: {}          43: {14}       102: {1,2,7}
     2: {1}         47: {15}       103: {27}
     3: {2}         53: {16}       105: {2,3,4}
     5: {3}         59: {17}       107: {28}
     7: {4}         61: {18}       109: {29}
    11: {5}         66: {1,2,5}    110: {1,3,5}
    13: {6}         67: {19}       113: {30}
    17: {7}         70: {1,3,4}    114: {1,2,8}
    19: {8}         71: {20}       127: {31}
    23: {9}         73: {21}       130: {1,3,6}
    29: {10}        78: {1,2,6}    131: {32}
    30: {1,2,3}     79: {22}       137: {33}
    31: {11}        83: {23}       138: {1,2,9}
    37: {12}        89: {24}       139: {34}
    41: {13}        97: {25}       149: {35}
    42: {1,2,4}    101: {26}       151: {36}

First condition alone is A005117 (squarefree).
Second condition alone is A100959 (non-semiprime).
The nonprime case is 1 followed by A350352.
Partitions of this type are counted by A365827, non-strict A058984.
A001358 lists semiprimes, squarefree A006881.
Cf. A000009, A004526, A008967, A078408, A365659.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]!=2&]

isok(k) = my(f=factor(k)); issquarefree(f) && (bigomega(f) != 2); \\ Michel Marcus, Oct 07 2023

Intersection of A005117 and A100959.

Complement of A001358 in A005117.

cp's OEIS Frontend

A282892 The difference between the number of partitions of n into odd parts (A000009) and the number of partitions of n into even parts (A035363).

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A300351 Triangle whose n-th row lists in order all Heinz numbers of integer partitions of n into odd parts.

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A346634 Number of strict odd-length integer partitions of 2n + 1.

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A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

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A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k*(2*k+1)) / Product_{j=1..2*k} (1 - x^j).

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A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

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A341448 Heinz numbers of integer partitions of type OO.

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A341449 Heinz numbers of integer partitions into odd parts > 1.

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A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k(2k+1)) / Product_{j=1..2*k} (1 - x^j).