# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a035363 Showing 1-1 of 1 %I A035363 #71 Sep 23 2023 12:11:08 %S A035363 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, %T A035363 0,176,0,231,0,297,0,385,0,490,0,627,0,792,0,1002,0,1255,0,1575,0, %U A035363 1958,0,2436,0,3010,0,3718,0,4565,0,5604,0,6842,0,8349,0,10143,0,12310,0 %N A035363 Number of partitions of n into even parts. %C A035363 Convolved with A036469 = A000070. - _Gary W. Adamson_, Jun 09 2009 %C A035363 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 %C A035363 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 %C A035363 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 %C A035363 From _Gus Wiseman_, May 22 2021: (Start) %C A035363 The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290. %C A035363 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: %C A035363 1 . 3 . 5 . 7 . 9 . B . D %C A035363 21 41 43 63 65 85 %C A035363 221 61 81 83 A3 %C A035363 421 441 A1 C1 %C A035363 2221 621 443 643 %C A035363 4221 641 661 %C A035363 22221 821 841 %C A035363 4421 A21 %C A035363 6221 4441 %C A035363 42221 6421 %C A035363 222221 8221 %C A035363 44221 %C A035363 62221 %C A035363 422221 %C A035363 2222221 %C A035363 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: %C A035363 (11) (22) (33) (44) (55) (66) %C A035363 (211) (321) (422) (532) (633) %C A035363 (3111) (431) (541) (642) %C A035363 (4211) (5221) (651) %C A035363 (41111) (5311) (6222) %C A035363 (52111) (6321) %C A035363 (511111) (6411) %C A035363 (62211) %C A035363 (63111) %C A035363 (621111) %C A035363 (6111111) %C A035363 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: %C A035363 (2) (22) (222) (2222) (22222) (222222) (2222222) %C A035363 (31) (321) (3221) (32221) (322221) (3222221) %C A035363 (411) (3311) (33211) (332211) (3322211) %C A035363 (4211) (42211) (333111) (3332111) %C A035363 (5111) (43111) (422211) (4222211) %C A035363 (52111) (432111) (4322111) %C A035363 (61111) (441111) (4331111) %C A035363 (522111) (4421111) %C A035363 (531111) (5222111) %C A035363 (621111) (5321111) %C A035363 (711111) (5411111) %C A035363 (6221111) %C A035363 (6311111) %C A035363 (7211111) %C A035363 (8111111) %C A035363 (End) %D A035363 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). %D A035363 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. %H A035363 Robert Price, Table of n, a(n) for n = 0..2001 %F A035363 G.f.: Product_{k even} 1/(1 - x^k). %F A035363 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 %F A035363 If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - _Omar E. Pol_, Nov 20 2009 %F A035363 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 %F A035363 a(n) = A096441(n) - A000009(n), n >= 1. - _Omar E. Pol_, Aug 16 2013 %F A035363 G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - _Ilya Gutkovskiy_, Aug 13 2018 %e A035363 From _Joerg Arndt_, Jun 11 2013: (Start) %e A035363 There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice: %e A035363 01: [ 1 1 1 1 1 2 2 1 1 1 1 1 ] %e A035363 02: [ 1 1 1 1 3 3 1 1 1 1 ] %e A035363 03: [ 1 1 1 4 4 1 1 1 ] %e A035363 04: [ 1 1 2 3 3 2 1 1 ] %e A035363 05: [ 1 1 5 5 1 1 ] %e A035363 06: [ 1 2 4 4 2 1 ] %e A035363 07: [ 1 6 6 1 ] %e A035363 08: [ 2 2 3 3 2 2 ] %e A035363 09: [ 2 5 5 2 ] %e A035363 10: [ 3 4 4 3 ] %e A035363 11: [ 7 7 ] %e A035363 There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times: %e A035363 01: [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] %e A035363 02: [ 1 1 1 1 1 2 2 1 1 1 1 1 ] %e A035363 03: [ 1 1 1 1 3 3 1 1 1 1 ] %e A035363 04: [ 1 1 1 2 2 2 2 1 1 1 ] %e A035363 05: [ 1 1 1 4 4 1 1 1 ] %e A035363 06: [ 1 1 2 3 3 2 1 1 ] %e A035363 07: [ 1 1 5 5 1 1 ] %e A035363 08: [ 1 2 2 2 2 2 2 1 ] %e A035363 09: [ 1 2 4 4 2 1 ] %e A035363 10: [ 1 3 3 3 3 1 ] %e A035363 11: [ 1 6 6 1 ] %e A035363 12: [ 2 2 3 3 2 2 ] %e A035363 13: [ 2 5 5 2 ] %e A035363 14: [ 3 4 4 3 ] %e A035363 15: [ 7 7 ] %e A035363 (End) %e A035363 a(8)=5 because we have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016 %e A035363 From _Gus Wiseman_, May 22 2021: (Start) %e A035363 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. %e A035363 () . (2) . (4) . (6) . (8) . (A) . (C) %e A035363 (22) (42) (44) (64) (66) %e A035363 (222) (62) (82) (84) %e A035363 (422) (442) (A2) %e A035363 (2222) (622) (444) %e A035363 (4222) (642) %e A035363 (22222) (822) %e A035363 (4422) %e A035363 (6222) %e A035363 (42222) %e A035363 (222222) %e A035363 (End) %p A035363 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 %p A035363 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 %p A035363 # Using the function EULER from Transforms (see link at the bottom of the page). %p A035363 [1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # _Peter Luschny_, Aug 19 2020 %p A035363 # next Maple program: %p A035363 a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)): %p A035363 seq(a(n), n=0..84); # _Alois P. Heinz_, Jun 22 2021 %t A035363 nmax = 50; s = Range[2, nmax, 2]; %t A035363 Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Aug 05 2020 *) %o A035363 (Python) %o A035363 from sympy import npartitions %o A035363 def A035363(n): return 0 if n&1 else npartitions(n>>1) # _Chai Wah Wu_, Sep 23 2023 %Y A035363 Bisection (even part) gives the partition numbers A000041. %Y A035363 Column k=0 of A103919, A264398. %Y A035363 Cf. A036469, A000070. %Y A035363 Cf. A135010, A138121. %Y A035363 Note: A-numbers of ranking sequences are in parentheses below. %Y A035363 The version for odd instead of even parts is A000009 (A066208). %Y A035363 The version for parts divisible by 3 instead of 2 is A035377. %Y A035363 The strict case is A035457. %Y A035363 The Heinz numbers of these partitions are given by A066207. %Y A035363 The ordered version (compositions) is A077957 prepended by (1,0). %Y A035363 This is column k = 2 of A168021. %Y A035363 The multiplicative version (factorizations) is A340785. %Y A035363 A000569 counts graphical partitions (A320922). %Y A035363 A004526 counts partitions of length 2 (A001358). %Y A035363 A025065 counts palindromic partitions (A265640). %Y A035363 A027187 counts partitions with even length/maximum (A028260/A244990). %Y A035363 A058696 counts partitions of even numbers (A300061). %Y A035363 A067661 counts strict partitions of even length (A030229). %Y A035363 A236913 counts partitions of even length and sum (A340784). %Y A035363 A340601 counts partitions of even rank (A340602). %Y A035363 The following count partitions of even length: %Y A035363 - A096373 cannot be partitioned into strict pairs (A320891). %Y A035363 - A338914 can be partitioned into strict pairs (A320911). %Y A035363 - A338915 cannot be partitioned into distinct pairs (A320892). %Y A035363 - A338916 can be partitioned into distinct pairs (A320912). %Y A035363 - A339559 cannot be partitioned into distinct strict pairs (A320894). %Y A035363 - A339560 can be partitioned into distinct strict pairs (A339561). %Y A035363 Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447. %K A035363 nonn,easy %O A035363 0,5 %A A035363 _Olivier Gérard_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE