A341447 -id:A341447 - cp's OEIS frontend

A035363 Number of partitions of n into even parts.

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

A087897 Number of partitions of n into odd parts greater than 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, 13, 15, 18, 20, 23, 27, 30, 34, 40, 44, 50, 58, 64, 73, 83, 92, 104, 118, 131, 147, 166, 184, 206, 232, 256, 286, 320, 354, 394, 439, 485, 538, 598, 660, 730, 809, 891, 984, 1088, 1196, 1318, 1454, 1596, 1756

Offset: 0

N. J. A. Sloane, Dec 04 2003

Also number of partitions of n into distinct parts which are not powers of 2.
Also number of partitions of n into distinct parts such that the two largest parts differ by 1.
Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006
Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007
In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013
Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014
From Gus Wiseman, Aug 22 2021: (Start)
Also the number of integer partitions of n - 1 whose parts cover an interval of positive integers starting with 2. These partitions are ranked by A339886. For example, the a(6) = 1 through a(16) = 5 partitions are:
  32  222  322  332   432   3322   3332   4332    4432    5432     43332
                2222  3222  22222  4322   33222   33322   33332    44322
                                   32222  222222  43222   43322    333222
                                                  322222  332222   432222
                                                          2222222  3222222
(End)

			1 + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + 3*x^13 + ...
q + q^73 + q^121 + q^145 + q^169 + q^193 + 2*q^217 + 2*q^241 + 2*q^265 + ...
a(10)=2 because we have [7,3] and [5,5].
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(22)=13 symmetric unimodal compositions of 22+3=25 where the maximal part appears three times:
01:  [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ]
03:  [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ]
04:  [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ]
05:  [ 1 1 1 2 5 5 5 2 1 1 1 ]
06:  [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ]
07:  [ 1 1 3 5 5 5 3 1 1 ]
08:  [ 1 1 7 7 7 1 1 ]
09:  [ 1 2 2 5 5 5 2 2 1 ]
10:  [ 1 4 5 5 5 4 1 ]
11:  [ 2 2 2 2 3 3 3 2 2 2 2 ]
12:  [ 2 3 5 5 5 3 2 ]
13:  [ 2 7 7 7 2 ]
(End)
From _Gus Wiseman_, Feb 16 2021: (Start)
The a(7) = 1 through a(19) = 8 partitions are the following (A..J = 10..19). The Heinz numbers of these partitions are given by A341449.
  7  53  9    55  B    75    D    77    F      97    H      99      J
         333  73  533  93    553  95    555    B5    755    B7      775
                       3333  733  B3    753    D3    773    D5      955
                                  5333  933    5533  953    F3      973
                                        33333  7333  B33    5553    B53
                                                     53333  7533    D33
                                                            9333    55333
                                                            333333  73333
(End)

J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from Alois P. Heinz)
C. Ballantine and M. Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1; doi.
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
Howard D. Grossman, Problem 228, Mathematics Magazine, 28 (1955), p. 160.
R. K. Guy, Two theorems on partitions, Math. Gaz., 42 (1958), 84-86. Math. Rev. 20 #3110.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

The ordered version is A000931.
Partitions with no ones are counted by A002865, ranked by A005408.
The even version is A035363, ranked by A066207.
The version for factorizations is A340101.
Partitions whose only even part is the smallest are counted by A341447.
The Heinz numbers of these partitions are given by A341449.
A000009 counts partitions into odd parts, ranked by A066208.
A025147 counts strict partitions with no 1's.
A025148 counts strict partitions with no 1's or 2's.
A026804 counts partitions whose smallest part is odd, ranked by A340932.
A027187 counts partitions with even length/maximum, ranks A028260/A244990.
A027193 counts partitions with odd length/maximum, ranks A026424/A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A058696 counts partitions of even numbers, ranked by A300061.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Cf. A000041, A003114, A039900, A160786, A257991/A257992, A264396, A300272, A339662, A339737, A339886.

a087897 = p [3,5..] where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

To get 128 terms: t4 := mul((1+x^(2^n)),n=0..7); t5 := mul((1+x^k),k=1..128): t6 := series(t5/t4,x,100); t7 := seriestolist(t6);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<3, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 11 2013

max = 65; f[x_] := Product[ 1/(1 - x^(2k+1)), {k, 1, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 16 2011, after Emeric Deutsch *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<3, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]] ]; a[n_] := b[n, n-1+Mod[n, 2]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* Vaclav Kotesovec, Dec 01 2015 *)
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&OddQ[Times@@#]&]],{n,0,30}] (* Gus Wiseman, Feb 16 2021 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - x) * eta(x^2 + A) / eta(x + A), n))} /* Michael Somos, Nov 13 2011 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A087897_T(n,k):
    if n==0: return 1
    if k<3 or n<0: return 0
    return A087897_T(n,k-2)+A087897_T(n-k,k)
def A087897(n): return A087897_T(n,n-(n&1^1)) # Chai Wah Wu, Sep 23 2023, after Alois P. Heinz

Expansion of q^(-1/24) * (1 - q) * eta(q^2) / eta(q) in powers of q.
Expansion of (1 - x) / chi(-x) in powers of x where chi() is a Ramanujan theta function.
G.f.: 1 + x^3 + x^5*(1 + x) + x^7*(1 + x)*(1 + x^2) + x^9*(1 + x)*(1 + x^2)*(1 + x^3) + ... [Glaisher 1876]. - Michael Somos, Jun 20 2012
G.f.: Product_{k >= 1} 1/(1-x^(2*k+1)).
G.f.: Product_{k >= 1, k not a power of 2} (1+x^k).
G.f.: Sum_{k >= 1} x^(3*k)/Product_{j = 1..k} (1 - x^(2*j)). - Emeric Deutsch, Mar 30 2006
a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (8 * 3^(3/4) * n^(5/4)) * (1 - (15*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (169*Pi^2/13824 + 385/384 + 315/(128*Pi^2))/n). - Vaclav Kotesovec, Aug 30 2015, extended Nov 04 2016
G.f.: 1/(1 - x^3) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 - x^3)*(1 - x^5)) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = ..., extending Deutsch's result dated Mar 30 2006. - Peter Bala, Jan 15 2021
G.f.: Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 2..2*n+1} (1 - x^k). (Set z = x^3 and q = x^2 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
a(2*n+1) = Sum{j>=1} A008284(n+1-j,2*j - 1) and a(2*n) = Sum{j>=1} A008284(n-j, 2*j). - Gregory L. Simay, Sep 22 2023

A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

Original entry on oeis.org

2, 5, 6, 11, 14, 17, 18, 23, 26, 31, 35, 38, 41, 42, 47, 54, 58, 59, 65, 67, 73, 74, 78, 83, 86, 95, 97, 98, 103, 106, 109, 114, 122, 126, 127, 137, 142, 143, 145, 149, 157, 158, 162, 167, 174, 178, 179, 182, 185, 191, 197, 202, 209, 211, 214, 215, 222, 226

Offset: 1

Gus Wiseman, Feb 12 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 odd prime index (counting multiplicity) is the smallest.

			The sequence of partitions together with their Heinz numbers begins:
      2: (1)         54: (2,2,2,1)    109: (29)
      5: (3)         58: (10,1)       114: (8,2,1)
      6: (2,1)       59: (17)         122: (18,1)
     11: (5)         65: (6,3)        126: (4,2,2,1)
     14: (4,1)       67: (19)         127: (31)
     17: (7)         73: (21)         137: (33)
     18: (2,2,1)     74: (12,1)       142: (20,1)
     23: (9)         78: (6,2,1)      143: (6,5)
     26: (6,1)       83: (23)         145: (10,3)
     31: (11)        86: (14,1)       149: (35)
     35: (4,3)       95: (8,3)        157: (37)
     38: (8,1)       97: (25)         158: (22,1)
     41: (13)        98: (4,4,1)      162: (2,2,2,2,1)
     42: (4,2,1)    103: (27)         167: (39)
     47: (15)       106: (16,1)       174: (10,2,1)

These partitions are counted by A035363 (shifted left once).
Terms of A340932 can be factored into elements of this sequence.
The even version is A341447.
A001222 counts prime factors.
A005408 lists odd numbers.
A026804 counts partitions whose smallest part is odd.
A027193 counts odd-length partitions, ranked by A026424.
A031368 lists odd-indexed primes.
A032742 selects largest proper divisor.
A055396 selects smallest prime index.
A056239 adds up prime indices.
A058695 counts partitions of odd numbers, ranked by A300063.
A061395 selects largest prime index.
A066207 lists numbers with all even prime indices.
A066208 lists numbers with all odd prime indices.
A112798 lists the prime indices of each positive integer.
A244991 lists numbers whose greatest prime index is odd.
A340932 lists numbers whose smallest prime index is odd.
Cf. A006141, A160786, A257991, A257992, A300272, A340604, A340607, A340854/A340855, A340933.

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

Also numbers n > 1 such that A055396(n) is odd and A032742(n) belongs to A066207.

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A035363 Number of partitions of n into even parts.

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A087897 Number of partitions of n into odd parts greater than 1.

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A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

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