A300061 -id:A300061 - cp's OEIS frontend

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A027187 Number of partitions of n into an even number of parts.

1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338, 73680, 86716, 102162, 119918

Offset: 0

Clark Kimberling

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]
Number of partitions of n+1 into an odd number of parts, the least being 1.
Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...
From _Gus Wiseman_, Jan 05 2021: (Start)
The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (2211)    (2221)    (71)
                              (3111)    (3211)    (2222)
                              (111111)  (4111)    (3221)
                                        (211111)  (3311)
                                                  (4211)
                                                  (5111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)
The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.
  (2)  (21)  (4)    (41)    (6)      (43)      (8)
             (22)   (221)   (42)     (61)      (44)
             (211)  (2111)  (222)    (421)     (62)
                            (411)    (2221)    (422)
                            (2211)   (4111)    (431)
                            (21111)  (22111)   (611)
                                     (211111)  (2222)
                                               (4211)
                                               (22211)
                                               (41111)
                                               (221111)
                                               (2111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 13.
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000701, A046682, A026838, A102462.
The Heinz numbers of these partitions are A028260.
The odd version is A027193.
The strict case is A067661.
The case of even sum as well as length is A236913 (the even bisection).
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A000009 counts partitions into odd parts, ranked by A066208.
A026805 counts partitions whose least part is even.
A072233 counts partitions by sum and length.
A101708 counts partitions of even positive rank.
Cf. A000700, A026424, A058696, A096373, A244990, A300061.

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)
a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

my(q='q+O('q^66)); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.
a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004
From Bill Gosper, Jun 25 2005: (Start)
G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3*q^4 + 3*q^5 + 6*q^6 + ...
  = Sum_{n >= 0} q^(2*n)/(q; q)_{2*n}
  = ((Product_{k >= 1} 1/(1-q^k)) + (Product_{k >= 1} 1/(1+q^k)))/2.
Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + ...
Then B(q) = Sum_{n >= 0} q^(2*n+1)/(q; q){2*n+1} = ((Product{k >= 1} 1/(1-q^k)) - (Product_{k >= 1} 1/(1+q^k)))/2.
Also we have the following identity involving 2 X 2 matrices:
Product_{k >= 1} [ 1/(1-q^(2*k)), q^k/(1-q^(2*k)) ; q^k/(1-q^(2*k)), 1/(1-q^(2*k)) ]
  = [ A(q), B(q) ; B(q), A(q) ]. (End)
a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006
Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006
G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006
a(n) = A338914(n) + A096373(n). - Gus Wiseman, Jan 06 2021

Offset changed to 0 by Michael Somos, Jul 24 2012

A025065 Number of palindromic partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296, 11732, 11732, 14742, 14742, 18460, 18460, 23025, 23025, 28629, 28629

Offset: 0

Clark Kimberling

nonn

That is, the number of partitions of n into parts which can be listed in palindromic order.
Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - T. D. Noe, Aug 05 2005
Also, partial sums of A035363.
Also number of partitions of n with at most one part occurring an odd number of times. - Reinhard Zumkeller, Dec 18 2013
The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - Clark Kimberling, Jan 20 2014
a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - Clark Kimberling, Apr 20 2014.
From Gus Wiseman, Nov 28 2018: (Start)
Also the number of integer partitions of n + 2 that are the vertex-degrees of some hypertree. For example, the a(6) = 7 partitions of 8 that are the vertex-degrees of some hypertree, together with a realizing hypertree are:
     (41111): {{1,2},{1,3},{1,4},{1,5}}
     (32111): {{1,2},{1,3},{1,4},{2,5}}
     (22211): {{1,2},{1,3},{2,4},{3,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
    (221111): {{1,2},{1,3},{2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}
(End)
Conjecture: a(n) is the length of maximal initial segment of A308355(n-1) that is identical to row n of A128628, for n >= 2. - Clark Kimberling, May 24 2019
From Gus Wiseman, May 21 2021: (Start)
The Heinz numbers of palindromic partitions are given by A265640. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
Also the number of integer partitions of n with a part greater than or equal to n/2. This is equivalent to Clark Kimberling's final comment above. The Heinz numbers of these partitions are given by A344414. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)     (8)
       (11)  (21)  (22)   (32)   (33)    (43)    (44)
                   (31)   (41)   (42)    (52)    (53)
                   (211)  (311)  (51)    (61)    (62)
                                 (321)   (421)   (71)
                                 (411)   (511)   (422)
                                 (3111)  (4111)  (431)
                                                 (521)
                                                 (611)
                                                 (4211)
                                                 (5111)
                                                 (41111)
Also the number of integer partitions of n with at least n/2 parts. The Heinz numbers of these partitions are given by A344296. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (21)   (22)    (221)    (222)     (2221)     (2222)
       (11)  (111)  (31)    (311)    (321)     (3211)     (3221)
                    (211)   (2111)   (411)     (4111)     (3311)
                    (1111)  (11111)  (2211)    (22111)    (4211)
                                     (3111)    (31111)    (5111)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			The partitions for the first few values of n are as follows:
n: partitions .......................... number
1: 1 ................................... 1
2: 2 11 ................................ 2
3: 3 111 ............................... 2
4: 4 22 121 1111 ....................... 4
5: 5 131 212 11111 ..................... 4
6: 6 141 33 222 1221 11211 111111 ...... 7
7: 7 151 313 11311 232 21112 1111111 ... 7
From _Reinhard Zumkeller_, Jan 23 2010: (Start)
Partitions into 1,2,4,6,... for the first values of n:
1: 1 ....................................... 1
2: 2 11 .................................... 2
3: 21 111 .................................. 2
4: 4 22 211 1111 ........................... 4
5: 41 221 2111 11111 ....................... 4
6: 6 42 4211 222 2211 21111 111111.......... 7
7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 101 terms from Reinhard Zumkeller that were corrected by _Georg Fischer_, Jan 20 2019)

Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010
Cf. A004526, A030019, A056503, A147878, A320921, A322136.
The bisections are both A000070.
The ordered version (palindromic compositions) is A016116.
The complement is counted by A233771 and A210249.
The case of palindromic prime signature is A242414.
Palindromic partitions are ranked by A265640, with complement A229153.
The case of palindromic plane trees is A319436.
The multiplicative version (palindromic factorizations) is A344417.
A000569 counts graphical partitions.
A027187 counts partitions of even length, ranked by A028260.
A035363 counts partitions into even parts, ranked by A066207.
A058696 counts partitions of even numbers, ranked by A300061.
A110618 counts partitions with length <= half sum, ranked by A344291.
Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.

a025065 = p (1:[2,4..]) 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

import Data.List (group)
a025065 = length . filter (<= 1) .
                   map (sum . map ((`mod` 2) . length) . group) . ps 1
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 18 2013

Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* Peter J. C. Moses, Jan 20 2014 *)
n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* Peter J. C. Moses, Jan 20 2014 *)
CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* Clark Kimberling, Mar 14 2014 *)

N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ Joerg Arndt, Mar 11 2014

a(n) = A000070(A004526(n)). - Reinhard Zumkeller, Jan 23 2010
G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [Joerg Arndt, Mar 11 2014]
a(2*k+2) = a(2*k) + A000041(k + 1). - David A. Corneth, May 29 2021
a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - Vaclav Kotesovec, Nov 16 2021

Edited by N. J. A. Sloane, Dec 29 2007

Prepended a(0)=1, added more terms, Joerg Arndt, Mar 11 2014

A035363 Number of partitions of n into even parts.

Original entry on oeis.org

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

A339846 Number of even-length factorizations of n into factors > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 12, 16, 24, 36, 48, 72, 96, 120:
  2*6  2*8      3*8      4*9      6*8      8*9      2*48         2*60
  3*4  4*4      4*6      6*6      2*24     2*36     3*32         3*40
       2*2*2*2  2*12     2*18     3*16     3*24     4*24         4*30
                2*2*2*3  3*12     4*12     4*18     6*16         5*24
                         2*2*3*3  2*2*2*6  6*12     8*12         6*20
                                  2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                           2*2*3*6  2*2*4*6      10*12
                                           2*3*3*4  2*3*4*4      2*2*5*6
                                                    2*2*2*12     2*3*4*5
                                                    2*2*2*2*2*3  2*2*2*15
                                                                 2*2*3*10

The case of set partitions (or n squarefree) is A024430.
The case of partitions (or prime powers) is A027187.
The ordered version is A174725, odd: A174726.
The odd-length factorizations are counted by A339890.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A316439 counts factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Cf. A002033, A007716, A027193, A050320, A058695, A074206, A236913, A320655, A320656, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, g(n$2, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

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],EvenQ@Length[#]&]],{n,100}]

A339846(n, m=n, e=1) = if(1==n, e, sumdiv(n, d, if((d>1)&&(d<=m), A339846(n/d, d, 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(n) + A339890(n) = A001055(n).

Data section extended up to a(105) by Antti Karttunen, Oct 22 2023

A000212 a(n) = floor(n^2/3).

Original entry on oeis.org

0, 0, 1, 3, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936

Offset: 0

N. J. A. Sloane

Let M_n be the n X n matrix of the following form: [3 2 1 0 0 0 0 0 0 0 / 2 3 2 1 0 0 0 0 0 0 / 1 2 3 2 1 0 0 0 0 0 / 0 1 2 3 2 1 0 0 0 0 / 0 0 1 2 3 2 1 0 0 0 / 0 0 0 1 2 3 2 1 0 0 / 0 0 0 0 1 2 3 2 1 0 / 0 0 0 0 0 1 2 3 2 1 / 0 0 0 0 0 0 1 2 3 2 / 0 0 0 0 0 0 0 1 2 3]. Then for n > 2 a(n) = det M_(n-2). - Benoit Cloitre, Jun 20 2002
Largest possible size for the directed Cayley graph on two generators having diameter n - 2. - Ralf Stephan, Apr 27 2003
It seems that for n >= 2, a(n) is the maximum number of non-overlapping 1 X 3 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's tool, see links. - Dmitry Kamenetsky, Aug 03 2009
Maximum number of edges in a K4-free graph with n vertices. - Yi Yang, May 23 2012
3a(n) + 1 = y^2 if n is not 0 mod 3 and 3a(n) = y^2 otherwise. - Jon Perry, Sep 10 2012
Apart from the initial term this is the elliptic troublemaker sequence R_n(1, 3) (also sequence R_n(2, 3)) in the notation of Stange (see Table 1, p. 16). For other elliptic troublemaker sequences R_n(a, b) see the cross references below. - Peter Bala, Aug 08 2013
The number of partitions of 2n into exactly 3 parts. - Colin Barker, Mar 22 2015
a(n-1) is the maximum number of non-overlapping triples (i,k), (i+1, k+1), (i+2, k+2) in an n X n matrix. Details: The triples are distributed along the main diagonal and 2*(n-1) other diagonals. Their maximum number is floor(n/3) + 2*Sum_{k = 1..n-1} floor(k/3) = floor((n-1)^2/3). - Gerhard Kirchner, Feb 04 2017
Conjecture:  a(n) is the number of intersection points of n cevians that cut a triangle into the maximum number of pieces (see A007980). - Anton Zakharov, May 07 2017
From Gus Wiseman, Oct 05 2020: (Start)
Also the number of unimodal triples (meaning the middle part is not strictly less than both of the other two) of positive integers summing to n + 1. The a(2) = 1 through a(6) = 12 triples are:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,2,4)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,3,3)
                    (2,2,1)  (1,4,1)  (1,4,2)
                    (3,1,1)  (2,2,2)  (1,5,1)
                             (2,3,1)  (2,2,3)
                             (3,2,1)  (2,3,2)
                             (4,1,1)  (2,4,1)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,2,1)
                                      (5,1,1)
(End)

			G.f. = x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 21*x^8 + 27*x^9 + 33*x^10 + ...
From _Gus Wiseman_, Oct 07 2020: (Start)
The a(2) = 1 through a(6) = 12 partitions of 2*n into exactly 3 parts (Barker) are the following. The Heinz numbers of these partitions are given by the intersection of A014612 (triples) and A300061 (even sum).
  (2,1,1)  (2,2,2)  (3,3,2)  (4,3,3)  (4,4,4)
           (3,2,1)  (4,2,2)  (4,4,2)  (5,4,3)
           (4,1,1)  (4,3,1)  (5,3,2)  (5,5,2)
                    (5,2,1)  (5,4,1)  (6,3,3)
                    (6,1,1)  (6,2,2)  (6,4,2)
                             (6,3,1)  (6,5,1)
                             (7,2,1)  (7,3,2)
                             (8,1,1)  (7,4,1)
                                      (8,2,2)
                                      (8,3,1)
                                      (9,2,1)
                                      (10,1,1)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, Generalized Schreier sets, linear recurrence relation, Turán graphs, arXiv:2112.14905 [math.CO], 2021.
Rafael Durbano Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.
Bakir Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.
Bakir Farhi, An Elementary Proof that any Natural Number can be Written as the Sum of Three Terms of the Sequence floor(n^2/3), Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.6.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Katherine E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051v3 [math.NT], 2011-2014.
C. K. Wong and Don Coppersmith, A combinatorial problem related to multimodule memory organizations, J. ACM 21 (1974), 392-402.
Anton Zakharov, Cevians.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000290, A007590 (= R_n(2,4)), A002620 (= R_n(1,2)), A118015, A056827, A118013.
Cf. A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A033440, A033441, A033442, A033443, A033444.
Cf. A001353 and A004523 (first differences). A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A238738. - Bruno Berselli, Apr 17 2015
Cf. A007980
Cf. A005408.
A000217(n-2) counts 3-part compositions.
A014612 ranks 3-part partitions, with strict case A007304.
A069905 counts the 3-part partitions.
A211540 counts strict 3-part partitions.
A337453 ranks strict 3-part compositions.
Cf. A056834, A130519, A056838, A056865.
A001399(n-6)*4 is the strict version.
A001523 counts unimodal compositions, with strict case A072706.
A001840(n-4) is the non-unimodal version.
A001399(n-6)*2 is the strict non-unimodal version.
A007052 counts unimodal patterns.
A115981 counts non-unimodal compositions, ranked by A335373.
A011782 counts unimodal permutations.
A335373 is the complement of a ranking sequence for unimodal compositions.
A337459 ranks these compositions, with complement A337460.
Cf. A069905, A220377, A332743, A337461, A337561, A337603, A337604.

[Floor(n^2 / 3): n in [0..50]]; // Vincenzo Librandi, May 08 2011

A000212:=(-1+z-2*z**2+z**3-2*z**4+z**5)/(z**2+z+1)/(z-1)**3; # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading 1.
A000212 := proc(n) option remember; `if`(n<4, [0,0,1,3][n+1], a(n-1)+a(n-3) -a(n-4)+2) end; # Peter Luschny, Nov 20 2011

Table[Quotient[n^2, 3], {n, 0, 59}] (* Michael Somos, Jan 22 2014 *)

{a(n) = n^2 \ 3}; /* Michael Somos, Sep 25 2006 */

def A000212(n): return n**2//3 # Chai Wah Wu, Jun 07 2022

G.f.: x^2*(1+x)/((1-x)^2*(1-x^3)). - Franklin T. Adams-Watters, Apr 01 2002
Euler transform of length 3 sequence [ 3, -1, 1]. - Michael Somos, Sep 25 2006
G.f.: x^2 * (1 - x^2) / ((1 - x)^3 * (1 - x^3)). a(-n) = a(n). - Michael Somos, Sep 25 2006
a(n) = Sum_{k = 0..n} A011655(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = a(n-1) + a(n-3) - a(n-4) + 2 for n >= 4. - Alexander Burstein, Nov 20 2011
a(n) = a(n-3) + A005408(n-2) for n >= 3. - Alexander Burstein, Feb 15 2013
a(n) = (n-1)^2 - a(n-1) - a(n-2) for n >= 2. - Richard R. Forberg, Jun 05 2013
Sum_{n >= 2} 1/a(n) = (27 + 6*sqrt(3)*Pi + 2*Pi^2)/36. - Enrique Pérez Herrero, Jun 29 2013
0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014
a(n) = Sum_{k = 1..n} k^2*A049347(n+2-k). - Mircea Merca, Feb 04 2014
a(n) = Sum_{i = 1..n+1} (ceiling(i/3) + floor(i/3) - 1). - Wesley Ivan Hurt, Jun 06 2014
a(n) = Sum_{j = 1..n} Sum_{i=1..n} ceiling((i+j-n-1)/3). - Wesley Ivan Hurt, Mar 12 2015
a(n) = Sum_{i = 1..n} floor(2*i/3). - Wesley Ivan Hurt, May 22 2017
a(-n) = a(n). - Paul Curtz, Jan 19 2020
a(n) = A001399(2*n - 3). - Gus Wiseman, Oct 07 2020
a(n) = (1/6)*(2*n^2 - 3 + gcd(n,3)). - Ridouane Oudra, Apr 15 2021
E.g.f.: (exp(x)*(-2 + 3*x*(1 + x)) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 24 2022
Sum_{n>=2} (-1)^n/a(n) = Pi/sqrt(3) - Pi^2/36 - 3/4. - Amiram Eldar, Dec 02 2022

Edited by Charles R Greathouse IV, Apr 19 2010

A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

Original entry on oeis.org

1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304

Offset: 1

N. J. A. Sloane

			Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). - _N. J. A. Sloane_, Jun 03 2012
From _Gus Wiseman_, Oct 27 2022: (Start)
The a(1) = 1 through a(4) = 14 partitions:
  (11)  (22)    (33)      (44)
        (211)   (321)     (422)
        (1111)  (2211)    (431)
                (3111)    (2222)
                (21111)   (3221)
                (111111)  (3311)
                          (4211)
                          (22211)
                          (32111)
                          (41111)
                          (221111)
                          (311111)
                          (2111111)
                          (11111111)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018.

Column m=2 of A213086.
Bisection of A276107.
Cf. A064914, A000041, A002220, A002221, A002222, A213074, A006827, A046663.
The strict version is A237258, ranked by A357854.
Ranked by A357976 = positions of nonzero terms in A357879.
A122768 counts distinct submultisets of partitions.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A108917, A235130, A237194, A300061.

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
b:= proc(n, i, s) option remember;
     `if`(i=1 and s<>{} or n in s, g(n, i), `if`(i<1 or s={}, 0,
      b(n, i-1, s)+ `if`(i>n, 0, b(n-i, i, map(x-> {`if`(x>n-i, NULL,
      max(x, n-i-x)), `if`(xn, NULL, max(x-i, n-x))}[], s)))))
    end:
a:= n-> b(2*n, n, {n}):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2012

Mathematica

b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *) primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; subptns[s_]:=primeMS/@Divisors[Times@@Prime/@s]; Table[Length[Select[IntegerPartitions[2n],MemberQ[Total/@subptns[#],n]&]],{n,10}] (* Gus Wiseman, Oct 27 2022 *)

Python

from itertools import combinations_with_replacement from sympy.utilities.iterables import partitions def A002219(n): return len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # Chai Wah Wu, Sep 20 2023

Formula

See A213074 for Metropolis and Stein's formulas.

a(n) = A000041(2*n) - A006827(n) = A000041(2*n) - A046663(2*n,n).

a(n) = A276107(2*n). - Max Alekseyev, Oct 17 2022

Extensions

Better description from Vladeta Jovovic, Mar 06 2000

More terms from Christian G. Bower, Oct 12 2001

Edited by N. J. A. Sloane, Jun 03 2012

More terms from Alois P. Heinz, Jul 10 2012

A195017 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} c_k*((-1)^(k-1)).

Original entry on oeis.org

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

Offset: 1

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History

Text

Internal format

Author

Clark Kimberling, Feb 06 2012

Keywords

sign
easy

Comments

Let p(n,x) be the completely additive polynomial-valued function such that p(1,x) = 0 and p(prime(n),x) = x^(n-1), like is defined in A206284 (although here we are not limited to just irreducible polynomials). Then a(n) is the value of the polynomial encoded in such a manner by n, when it is evaluated at x=-1. - The original definition rewritten and clarified by Antti Karttunen, Oct 03 2018

Positions of 0 give the values of n for which the polynomial p(n,x) is divisible by x+1. For related sequences, see the Mathematica section.

Also the number of odd prime indices of n minus the number of even prime indices of n (both counted with multiplicity), where 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. - Gus Wiseman, Oct 24 2023

Examples

The sequence can be read from a list of the polynomials: p(n,x) with x = -1, gives a(n) ------------------------------------------ p(1,x) = 0 0 p(2,x) = 1x^0 1 p(3,x) = x -1 p(4,x) = 2x^0 2 p(5,x) = x^2 1 p(6,x) = 1+x 0 p(7,x) = x^3 -1 p(8,x) = 3x^0 3 p(9,x) = 2x -2 p(10,x) = x^2 + 1 2. (The list runs through all the polynomials whose coefficients are nonnegative integers.)

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A206284, A277322, A284010.

For other evaluation functions of such encoded polynomials, see A001222, A048675, A056239, A090880, A248663.

Cf. A006093, A036689, A078437.

Zeros are A325698, distinct A325700.

For sum instead of count we have A366749 = A366531 - A366528.

A000009 counts partitions into odd parts, ranked by A066208.

A035363 counts partitions into even parts, ranked by A066207.

A112798 lists prime indices, reverse A296150, sum A056239.

A257991 counts odd prime indices, even A257992.

A300061 lists numbers with even sum of prime indices, odd A300063.

Cf. A019507, A106529, A239241, A239261, A369180.

Programs

Mathematica

b[n_] := Table[x^k, {k, 0, n}]; f[n_] := f[n] = FactorInteger[n]; z = 200; t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]] == Prime[k], f[n][[m, 2]], 0]; u = Table[Apply[Plus, Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1, Length[f[n]]}]], {n, 1, z}]; p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]] Table[p[n, x] /. x -> 0, {n, 1, z/2}] (* A007814 *) Table[p[2 n, x] /. x -> 0, {n, 1, z/2}] (* A001511 *) Table[p[n, x] /. x -> 1, {n, 1, z}] (* A001222 *) Table[p[n, x] /. x -> 2, {n, 1, z}] (* A048675 *) Table[p[n, x] /. x -> 3, {n, 1, z}] (* A090880 *) Table[p[n, x] /. x -> -1, {n, 1, z}] (* A195017 *) z = 100; Sum[-(-1)^k IntegerExponent[Range[z], Prime[k]], {k, 1, PrimePi[z]}] (* Friedjof Tellkamp, Aug 05 2024 *)

PARI

A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * (-1)^(1+primepi(f[i,1])))); } \\ Antti Karttunen, Oct 03 2018

Formula

Totally additive with a(p^e) = e * (-1)^(1+PrimePi(p)), where PrimePi(n) = A000720(n). - Antti Karttunen, Oct 03 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} = (-1)^(primepi(p)+1)/(p-1) = Sum_{k>=1} (-1)^(k+1)/A006093(k) = A078437 + Sum_{k>=1} (-1)^(k+1)/A036689(k) = 0.6339266524059... . - Amiram Eldar, Sep 29 2023

a(n) = A257991(n) - A257992(n). - Gus Wiseman, Oct 24 2023

a(n) = -Sum_{k=1..pi(n)} (-1)^k * valuation(n, prime(k)). - Friedjof Tellkamp, Aug 05 2024

Extensions

More terms, name changed and example-section edited by Antti Karttunen, Oct 03 2018

A067661 Number of partitions of n into distinct parts such that number of parts is even.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3959, 4404

Offset: 0

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Naohiro Nomoto, Feb 23 2002

Keywords

easy
nonn

Comments

Ramanujan theta functions: phi(q) (A000122), chi(q) (A000700).

Examples

G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + ... From _Gus Wiseman_, Jan 08 2021: (Start) The a(3) = 1 through a(14) = 11 partitions (A-D = 10..13): 21 31 32 42 43 53 54 64 65 75 76 86 41 51 52 62 63 73 74 84 85 95 61 71 72 82 83 93 94 A4 81 91 92 A2 A3 B3 4321 A1 B1 B2 C2 5321 5421 C1 D1 6321 5431 5432 6421 6431 7321 6521 7421 8321 (End)

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (2).

Links

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

Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff

Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_e(n).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Dominates A000009.

Numbers with these strict partitions as binary indices are A001969.

The non-strict case is A027187, ranked by A028260.

The Heinz numbers of these partitions are A030229.

The odd version is A067659, ranked by A030059.

The version for rank is A117192, with positive case A101708.

Other cases of even length:

- A024430 counts set partitions of even length.

- A034008 counts compositions of even length.

- A052841 counts ordered set partitions of even length.

- A174725 counts ordered factorizations of even length.

- A332305 counts strict compositions of even length

- A339846 counts factorizations of even length.

A008289 counts strict partitions by sum and length.

A026805 counts partitions whose least part is even.

Cf. A000700, A027193, A058696, A072233, A096373, A236913, A244990, A300061.

Programs

Maple

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(n$2, 1): seq(a(n), n=0..80); # Alois P. Heinz, Apr 01 2014

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *) a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] + QPochhammer[ x]) / 2, {x, 0, n}]; (* Michael Somos, May 06 2015 *) Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 08 2021 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A) + eta(x + A)) / 2, n))}; /* Michael Somos, Feb 14 2006 */

PARI

N=66; q='q+O('q^N); S=1+2*sqrtint(N); gf=sum(n=0, S, (n%2==0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ); Vec(gf) \\ Joerg Arndt, Apr 01 2014

Formula

G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^3 + q^4 + 2 q^5 + 2 q^6 + 3 q^7 + ... = Sum_{n >= 0} q^(n(2n+1))/(q; q){2n} [_Bill Gosper, Jun 25 2005]

Also, let B(q) = Sum_{n >= 0} A067659(n) q^n = q + q^2 + q^3 + q^4 + q^5 + 2 q^6 + ... Then B(q) = Sum_{n >= 0} q^((n+1)(2n+1))/(q; q)_{2n+1}.

Also we have the following identity involving 2 X 2 matrices:

Prod_{k >= 1} [ 1, q^k; q^k, 1 ] = [ A(q), B(q); B(q), A(q) ] [Bill Gosper, Jun 25 2005]

a(n) = (A000009(n)+A010815(n))/2. - Vladeta Jovovic, Feb 24 2002

Expansion of (1 + phi(-x)) / (2*chi(-x)) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006

a(n) + A067659(n) = A000009(n). - R. J. Mathar, Jun 18 2016

a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 24 2018

A000009(n) = a(n) + A067659(n). - Gus Wiseman, Jan 09 2021

From Peter Bala, Feb 05 2021: (Start)

G.f.: A(x) = (1/2)*((Product_{n >= 0} 1 + x^n) + (Product_{n >= 0} 1 - x^n)).

Let B(x) denote the g.f. of A067659. Then

A(x)^2 - B(x)^2 = A(x^2) - B(x^2) = Product_{n >= 1} 1 - x^(2*n) = Sum_{n in Z} (-1)^n*x^(n*(3*n+1)).

A(x) + B(x) is the g.f. of A000009.

1/(A(x) - B(x)) is the g.f. of A000041.

(A(x) + B(x))/(A(x) - B(x)) is the g.f. of A015128.

A(x)/(A(x) + B(x)) = Sum_{n >= 0} (-1)^n*x^n^2 = (1 + theta_3(-x))/2.

B(x)/(A(x) - B(x)) is the g.f. of A014968.

A(x)/(A(x^2) - B(x^2)) is the g.f. of A027187.

B(x)/(A(x^2) - B(x^2)) is the g.f. of A027193. (End)

A300063 Heinz numbers of integer partitions of odd numbers.

Original entry on oeis.org

2, 5, 6, 8, 11, 14, 15, 17, 18, 20, 23, 24, 26, 31, 32, 33, 35, 38, 41, 42, 44, 45, 47, 50, 51, 54, 56, 58, 59, 60, 65, 67, 68, 69, 72, 73, 74, 77, 78, 80, 83, 86, 92, 93, 95, 96, 97, 98, 99, 103, 104, 105, 106, 109, 110, 114, 119, 122, 123, 124, 125, 126, 127

Offset: 1

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Author

Gus Wiseman, Feb 23 2018

Keywords

nonn

Comments

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

Examples

15 is the Heinz number of (3,2), which has odd weight, so 15 belongs to the sequence. Sequence of odd-weight partitions begins: (1) (3) (2,1) (1,1,1) (5) (4,1) (3,2) (7) (2,2,1) (3,1,1) (9) (2,1,1,1) (6,1).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A300061.

Cf. A000041, A000720, A001222, A056239, A063834, A112798, A215366, A296150, A299757, A300056, A300060, A304662.

Programs

Maple

a:= proc(n) option remember; local k; for k from 1+ `if`(n=1, 0, a(n-1)) while add(numtheory[pi] (i[1])*i[2], i=ifactors(k)[2])::even do od; k end: seq(a(n), n=1..100); # Alois P. Heinz, May 22 2018

Mathematica

Select[Range[200],OddQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&]

A236913 Number of partitions of 2n of type EE (see Comments).

Original entry on oeis.org

1, 1, 3, 6, 12, 22, 40, 69, 118, 195, 317, 505, 793, 1224, 1867, 2811, 4186, 6168, 9005, 13026, 18692, 26613, 37619, 52815, 73680, 102162, 140853, 193144, 263490, 357699, 483338, 650196, 870953, 1161916, 1544048, 2044188, 2696627, 3545015, 4644850, 6066425

Offset: 0

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Author

Clark Kimberling, Feb 01 2014

Keywords

nonn
easy

Comments

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.

Examples

The partitions of 4 of type EE are [3,1], [2,2], [1,1,1,1], so that a(2) = 3. type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11 EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0 OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29 EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0 OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27 From _Gus Wiseman_, Feb 09 2021: (Start) This sequence counts even-length partitions of even numbers, which have Heinz numbers given by A340784. For example, the a(0) = 1 through a(4) = 12 partitions are: () (11) (22) (33) (44) (31) (42) (53) (1111) (51) (62) (2211) (71) (3111) (2222) (111111) (3221) (3311) (4211) (5111) (221111) (311111) (11111111) (End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A027193, A236559, A236914.

Note: A-numbers of ranking sequences are in parentheses below.

The ordered version is A000302.

The case of odd-length partitions of odd numbers is A160786 (A340931).

The Heinz numbers of these partitions are (A340784).

A027187 counts partitions of even length/maximum (A028260/A244990).

A034008 counts compositions of even length.

A035363 counts partitions into even parts (A066207).

A047993 counts balanced partitions (A106529).

A058695 counts partitions of odd numbers (A300063).

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A072233 counts partitions by sum and length.

A339846 counts factorizations of even length.

A340601 counts partitions of even rank (A340602).

A340785 counts factorizations into even factors.

A340786 counts even-length factorizations into even factors.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p-> `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]], [p[2], p[1], p[4], p[3]]))(b(n-i, i))))) end: a:= n-> b(2*n$2)[1]: seq(a(n), n=0..40); # Alois P. Heinz, Feb 16 2014

Mathematica

z = 25; m1 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &, OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]]; m2 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &, OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]]; m3 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &, OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &, OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]]; m1 (* A236559, type EO*) m2 (* A160786, type OE*) m3 (* A236913, type EE*) m4 (* A236914, type OO*) (* Peter J. C. Moses, Feb 03 2014 *) b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[i > n, {0, 0, 0, 0}, Function[p, If[Mod[i, 2] == 0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n - i, i]]]]]; a[n_] := b[2*n, 2*n][[1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 27 2015, after Alois P. Heinz *) Table[Length[Select[IntegerPartitions[2n],EvenQ[Length[#]]&]],{n,0,15}] (* Gus Wiseman, Feb 09 2021 *)

Extensions

More terms from Alois P. Heinz, Feb 16 2014

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A027187 Number of partitions of n into an even number of parts.

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A025065 Number of palindromic partitions of n.

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A035363 Number of partitions of n into even parts.

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A339846 Number of even-length factorizations of n into factors > 1.

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A000212 a(n) = floor(n^2/3).

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A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

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