A026805 -id:A026805 - cp's OEIS frontend

A026832 Number of partitions of n into distinct parts, the least being odd.

0, 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 8, 10, 12, 14, 18, 21, 24, 30, 36, 42, 50, 58, 68, 80, 93, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372, 3766, 4202, 4682

Offset: 0

Clark Kimberling

Fine's numbers L(n).

Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1], [2,2,2,1], [2,1,1,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 29 2006

			a(7)=4 because we have [7], [6,1], [4,3] and [4,2,1].

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A026804, A026805, A026807, A026833, A092265, A096661, A097042.

Cf. A000009, A096765.

a026832 n = p 1 n where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
-- Reinhard Zumkeller, Jun 14 2012

g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

mx=53; Rest[CoefficientList[Series[Sum[x^(2*k-1) Product[1+x^j, {j, 2*k, mx}], {k, mx}], {x, 0, mx}], x]]  (* Jean-François Alcover, Apr 05 2011, after Emeric Deutsch *)
Join[{0},Table[Length[Select[IntegerPartitions[n],OddQ[#[[-1]]]&&Max[Tally[#][[All,2]]] == 1&]],{n,60}]] (* Harvey P. Dale, May 14 2022 *)

G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 26 2003
G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - Vladeta Jovovic, Aug 10 2004
(1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
G.f.: Sum_{k>=1} x^(2k-1)*Product_{j>=2k} (1 + x^j). - Emeric Deutsch, Mar 29 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 09 2019

More terms from Emeric Deutsch, Mar 29 2006

a(0)=0 prepended by Alois P. Heinz, Feb 01 2019

A103419 Number of compositions of n in which the least part is odd.

Original entry on oeis.org

1, 1, 4, 6, 14, 28, 59, 117, 239, 484, 980, 1973, 3973, 7989, 16054, 32227, 64653, 129628, 259787, 520440, 1042305, 2086938, 4177680, 8361557, 16733221, 33482909, 66992641, 134028938, 268128902, 536373288, 1072934271, 2146173471, 4292842170, 8586488355

Offset: 1

Vladeta Jovovic, Feb 04 2005

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A103420-A103422, A027187, A027193, A026804, A026805.

b:= proc(n, i) option remember; `if`(n=0, irem(i, 2), add(
      (t-> b(t, min(i, j, `if`(t>0, t, j))))(n-j), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 26 2015

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n - 1)/((1 - x - x^(2n))*(1 - x - x^(2n - 1))), {n, 35}]], {x, 0, 35}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

G.f.: Sum((1-x)^2*x^(2*n-1)/((1-x-x^(2*n))*(1-x-x^(2*n-1))), n=1..infinity).
G.f.: Sum(x^k/((1-x)^k*(1+x^k)),k=1..infinity). - Vladeta Jovovic, Mar 02 2008
a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014

More terms from Robert G. Wilson v, Feb 05 2005

A103420 Number of compositions of n in which the least part is even.

Original entry on oeis.org

0, 1, 0, 2, 2, 4, 5, 11, 17, 28, 44, 75, 123, 203, 330, 541, 883, 1444, 2357, 3848, 6271, 10214, 16624, 27051, 43995, 71523, 116223, 188790, 306554, 497624, 807553, 1310177, 2125126, 3446237, 5587517, 9057611, 14680337, 23789891, 38546834, 62449682, 101163024

Offset: 1

Vladeta Jovovic, Feb 04 2005

Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 1000 terms from Robert Israel)

Cf. A103419, A103421, A103422, A027187, A027193, A026804, A026805.

N:= 50: # for a(1) .. a(N)
G:= add(x^(2*n)/((1-x)^n*(1+x^n)),n=1..N/2):
S:= series(G,x,N+1):
[seq(coeff(S,x,i),i=1..N)]; # Robert Israel, Oct 23 2024
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1-
      irem(m, 2), add(b(n-j, min(m, j)), j=1..n))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=1..42);  # Alois P. Heinz, Oct 23 2024

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n)/((1 - x - x^(2n))*(1 - x - x^(2n + 1))), {n, 40}]], {x, 0, 40}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

G.f.: Sum((1-x)^2*x^(2*n)/((1-x-x^(2*n))*(1-x-x^(2*n+1))), n=1..infinity).
G.f.: Sum(x^(2*n)/((1-x)^n*(1+x^n)),n=1..infinity). - Vladeta Jovovic, Mar 02 2008
a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-1). - Vaclav Kotesovec, May 01 2014

More terms from Robert G. Wilson v, Feb 05 2005

A103422 Number of compositions of n in which the greatest part is even.

Original entry on oeis.org

0, 1, 2, 5, 9, 18, 34, 66, 127, 249, 490, 972, 1936, 3874, 7772, 15623, 31439, 63308, 127506, 256782, 516970, 1040340, 2092450, 4206146, 8449953, 16965459, 34042784, 68272206, 136847328, 274168858, 549042730, 1099050180, 2199222960

Offset: 1

Vladeta Jovovic, Feb 04 2005

Cf. A103419, A103420, A103421, A027187, A027193, A026804, A026805.

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n)/((1 - 2x + x^(2n))*(1 - 2x + x^(2n + 1))), {n, 35}]], {x, 0, 35}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

G.f.: Sum((1-x)^2*x^(2*n)/((1-2*x+x^(2*n))*(1-2*x+x^(2*n+1))), n=1..infinity).

More terms from Robert G. Wilson v, Feb 05 2005

A341447 Heinz numbers of integer partitions whose only even part is the smallest.

Original entry on oeis.org

3, 7, 13, 15, 19, 29, 33, 37, 43, 51, 53, 61, 69, 71, 75, 77, 79, 89, 93, 101, 107, 113, 119, 123, 131, 139, 141, 151, 161, 163, 165, 173, 177, 181, 193, 199, 201, 217, 219, 221, 223, 229, 239, 249, 251, 255, 263, 271, 281, 287, 291, 293, 299, 309, 311, 317

Offset: 1

Gus Wiseman, Feb 13 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only even prime index (counting multiplicity) is the smallest.

			The sequence of partitions together with their Heinz numbers begins:
      3: (2)         77: (5,4)     165: (5,3,2)
      7: (4)         79: (22)      173: (40)
     13: (6)         89: (24)      177: (17,2)
     15: (3,2)       93: (11,2)    181: (42)
     19: (8)        101: (26)      193: (44)
     29: (10)       107: (28)      199: (46)
     33: (5,2)      113: (30)      201: (19,2)
     37: (12)       119: (7,4)     217: (11,4)
     43: (14)       123: (13,2)    219: (21,2)
     51: (7,2)      131: (32)      221: (7,6)
     53: (16)       139: (34)      223: (48)
     61: (18)       141: (15,2)    229: (50)
     69: (9,2)      151: (36)      239: (52)
     71: (20)       161: (9,4)     249: (23,2)
     75: (3,3,2)    163: (38)      251: (54)

These partitions are counted by A087897, shifted left once.
Terms of A340933 can be factored into elements of this sequence.
The odd version is A341446.
A000009 counts partitions into odd parts, ranked by A066208.
A001222 counts prime factors.
A005843 lists even numbers.
A026804 counts partitions whose least part is odd, ranked by A340932.
A026805 counts partitions whose least part is even, ranked by A340933.
A027187 counts partitions with even length/max, ranked by A028260/A244990.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058696 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A066207 lists numbers with all even prime indices.
A112798 lists the prime indices of each positive integer.
Cf. A026804, A035363, A036554, A160786, A244991, A257991, A257992, A300272, A300063, A340854/A340855.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],EvenQ[First[primeMS[#]]]&&And@@OddQ[Rest[primeMS[#]]]&]

A363094 Number of partitions of n whose least part is a multiple of 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 1, 3, 2, 3, 6, 6, 7, 11, 11, 14, 21, 24, 29, 38, 44, 54, 69, 81, 98, 123, 144, 174, 213, 253, 300, 363, 427, 508, 608, 716, 846, 1004, 1176, 1384, 1631, 1908, 2230, 2616, 3046, 3553, 4143, 4813, 5586, 6492, 7509, 8693, 10057, 11608, 13383, 15435, 17753, 20418, 23463, 26923, 30864

Offset: 1

Seiichi Manyama, May 19 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026805, A363095, A363096.

Cf. A363045.

nmax = 60; Rest[CoefficientList[Series[Sum[x^(3*k)/QPochhammer[x^(3*k), x], {k, 1, nmax/3}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

my(N=70, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=3*k, N, 1-x^j))))

G.f.: Sum_{k>=1} x^(3*k)/Product_{j>=3*k} (1-x^j).

a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (4 * 3^(3/2) * n^2) * (1 - (3*sqrt(6)/Pi + 109*Pi*sqrt(6)/144)/sqrt(n)). - Vaclav Kotesovec, May 21 2023

A363095 Number of partitions of n whose least part is a multiple of 4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 1, 3, 2, 3, 3, 7, 6, 8, 9, 13, 13, 17, 19, 28, 30, 38, 43, 56, 62, 76, 87, 110, 124, 151, 173, 211, 241, 289, 332, 399, 456, 539, 620, 733, 838, 983, 1127, 1322, 1513, 1761, 2016, 2343, 2677, 3096, 3536, 4083, 4655, 5355, 6101, 7005, 7969, 9124, 10370, 11856, 13453, 15340

Offset: 1

Seiichi Manyama, May 19 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026805, A363094, A363096.

Cf. A363046.

nmax = 60; Rest[CoefficientList[Series[Sum[x^(4*k)/QPochhammer[x^(4*k), x], {k, 1, nmax/4}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

my(N=70, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=4*k, N, 1-x^j))))

G.f.: Sum_{k>=1} x^(4*k)/Product_{j>=4*k} (1-x^j).

a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(5/2)) * (1 - (5*sqrt(6)/Pi + 169*Pi*sqrt(6)/144)/sqrt(n)). - Vaclav Kotesovec, May 21 2023

A363096 Number of partitions of n whose least part is a multiple of 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 1, 3, 2, 3, 3, 4, 7, 7, 8, 10, 11, 15, 16, 19, 22, 27, 34, 39, 46, 54, 63, 76, 86, 101, 117, 136, 161, 186, 214, 249, 287, 335, 384, 445, 509, 588, 677, 776, 888, 1020, 1163, 1334, 1519, 1735, 1975, 2253, 2564, 2917, 3312, 3762, 4265, 4842, 5477, 6203, 7012, 7928

Offset: 1

Seiichi Manyama, May 19 2023

nonn

In general, for m > 0, if g.f. = Sum_{k>=1} x^(m*k)/Product_{j>=m*k} (1-x^j), then a(n) ~ Pi^(m-1) * (m-1)! * exp(Pi*sqrt(2*n/3)) / (2^(3/2) * 6^(m/2) * n^((m+1)/2)) * (1 - (m*(m+1)/(4*Pi) + (6*m^2 + 18*m + 1 + c)*Pi/144)/sqrt(n/6)), where c = 0 for m > 1 and c = -24 for m = 1. - Vaclav Kotesovec, May 21 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026805, A363094, A363095.

Cf. A363047.

nmax = 60; Rest[CoefficientList[Series[Sum[x^(5*k)/QPochhammer[x^(5*k), x], {k, 1, nmax/5}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

my(N=70, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=5*k, N, 1-x^j))))

G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=5*k} (1-x^j).

a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (2*3^(3/2)*n^3) * (1 - (15*sqrt(6)/(2*Pi) + 241*Pi*sqrt(6)/144) / sqrt(n)). - Vaclav Kotesovec, May 21 2023

A103421 Number of compositions of n in which the greatest part is odd.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 30, 62, 129, 263, 534, 1076, 2160, 4318, 8612, 17145, 34097, 67764, 134638, 267506, 531606, 1056812, 2101854, 4182462, 8327263, 16588973, 33066080, 65945522, 131588128, 262702054, 524699094, 1048433468, 2095744336

Offset: 1

Vladeta Jovovic, Feb 04 2005

Cf. A103419, A103420, A103422, A027187, A027193, A026804, A026805.

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n - 1)/((1 - 2x + x^(2n - 1))*(1 - 2x + x^(2n))), {n, 35}]], {x, 0, 35}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

G.f.: Sum((1-x)^2*x^(2*n-1)/((1-2*x+x^(2*n-1))*(1-2*x+x^(2*n))), n=1..infinity).

a(n) + A103422(n) = 2^(n-1). - R. J. Mathar, Mar 24 2018

More terms from Robert G. Wilson v, Feb 05 2005

cp's OEIS Frontend

A026832 Number of partitions of n into distinct parts, the least being odd.

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A103419 Number of compositions of n in which the least part is odd.

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A103420 Number of compositions of n in which the least part is even.

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A103422 Number of compositions of n in which the greatest part is even.

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A341447 Heinz numbers of integer partitions whose only even part is the smallest.

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A363094 Number of partitions of n whose least part is a multiple of 3.

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A363095 Number of partitions of n whose least part is a multiple of 4.

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A363096 Number of partitions of n whose least part is a multiple of 5.

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