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%I A096441 #36 Jan 16 2022 11:08:51
%S A096441 1,2,2,4,3,7,5,11,8,17,12,26,18,37,27,54,38,76,54,106,76,145,104,199,
%T A096441 142,266,192,357,256,472,340,621,448,809,585,1053,760,1354,982,1740,
%U A096441 1260,2218,1610,2818,2048,3559,2590,4485,3264,5616,4097,7018,5120,8728,6378
%N A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.
%C A096441 Number of partitions of n such that all differences between successive parts are even, see example. [_Joerg Arndt_, Dec 27 2012]
%C A096441 Number of partitions of n where either all parts are odd or all parts are even. - _Omar E. Pol_, Aug 16 2013
%C A096441 From _Gus Wiseman_, Jan 13 2022: (Start)
%C A096441 Also the number of integer partitions of n with all even multiplicities (or run-lengths) except possibly the first. These are the conjugates of the partitions described by Joerg Arndt above. For example, the a(1) = 1 through a(8) = 11 partitions are:
%C A096441   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%C A096441        (11)  (111)  (22)    (311)    (33)      (322)      (44)
%C A096441                     (211)   (11111)  (222)     (511)      (422)
%C A096441                     (1111)           (411)     (31111)    (611)
%C A096441                                      (2211)    (1111111)  (2222)
%C A096441                                      (21111)              (3311)
%C A096441                                      (111111)             (22211)
%C A096441                                                           (41111)
%C A096441                                                           (221111)
%C A096441                                                           (2111111)
%C A096441                                                           (11111111)
%C A096441 (End)
%D A096441 A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras. Lie groups and invariant theory, 85-104, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 2005.
%H A096441 Alois P. Heinz, <a href="/A096441/b096441.txt">Table of n, a(n) for n = 1..1000</a>
%H A096441 Karin Baur and Nolan Wallach, <a href="http://dx.doi.org/10.1090/S1088-4165-05-00262-1">Nice parabolic subalgebras of reductive Lie algebras</a>, Represent. Theory 9 (2005), 1-29.
%H A096441 A. G. Elashvili and V. G. Kac, <a href="http://arxiv.org/abs/math-ph/0312030">Classification of good gradings of simple Lie algebras</a>, arXiv:math-ph/0312030, 2002-2004.
%F A096441 G.f.: sum(j>=1, q^j * (1-q^j)/prod(i=1..j, 1-q^(2*i) ) ).
%F A096441 G.f.: F + G - 2, where F = prod(j>=1, 1/(1-q^(2*j) ), G = prod(j>=0, 1/(1-q^(2*j+1)) ).
%F A096441 a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - _Vladeta Jovovic_, Aug 11 2004
%F A096441 a(n) = A000009(n) + A035363(n) = A000041(n) - A006477(n). - _Omar E. Pol_, Aug 16 2013
%e A096441 From _Joerg Arndt_, Dec 27 2012: (Start)
%e A096441 There are a(10)=17 partitions of 10 where all differences between successive parts are even:
%e A096441 [ 1]  [ 1 1 1 1 1 1 1 1 1 1 ]
%e A096441 [ 2]  [ 2 2 2 2 2 ]
%e A096441 [ 3]  [ 3 1 1 1 1 1 1 1 ]
%e A096441 [ 4]  [ 3 3 1 1 1 1 ]
%e A096441 [ 5]  [ 3 3 3 1 ]
%e A096441 [ 6]  [ 4 2 2 2 ]
%e A096441 [ 7]  [ 4 4 2 ]
%e A096441 [ 8]  [ 5 1 1 1 1 1 ]
%e A096441 [ 9]  [ 5 3 1 1 ]
%e A096441 [10]  [ 5 5 ]
%e A096441 [11]  [ 6 2 2 ]
%e A096441 [12]  [ 6 4 ]
%e A096441 [13]  [ 7 1 1 1 ]
%e A096441 [14]  [ 7 3 ]
%e A096441 [15]  [ 8 2 ]
%e A096441 [16]  [ 9 1 ]
%e A096441 [17]  [ 10 ]
%e A096441 (End)
%p A096441 b:= proc(n, i) option remember; `if`(i>n, 0,
%p A096441       `if`(irem(n, i)=0, 1, 0) +add(`if`(irem(j, 2)=0,
%p A096441        b(n-i*j, i+1), 0), j=0..n/i))
%p A096441     end:
%p A096441 a:= n-> b(n, 1):
%p A096441 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 26 2014
%t A096441 (* The following Mathematica program first generates all of the palindromic, unimodal compositions of n and then counts them. *)
%t A096441 Pal[n_] := Block[{i, j, k, m, Q, L}, If[n == 1, Return[{{1}}]]; If[n == 2, Return[{{1, 1}, {2}}]]; L = {{n}}; If[Mod[n, 2] == 0, L = Append[L, {n/2, n/2}]]; For[i = 1, i < n, i++, Q = Pal[n - 2i]; m = Length[Q]; For[j = 1, j <= m, j++, If[i <= Q[[j, 1]], L = Append[L, Append[Prepend[Q[[j]], i], i]]]]]; L] NoPal[n_] := Length[Pal[n]]
%t A096441 a[n_] := PartitionsQ[n] + If[EvenQ[n], PartitionsP[n/2], 0]; Table[a[n], {n, 1, 55}] (* _Jean-François Alcover_, Mar 17 2014, after _Vladeta Jovovic_ *)
%t A096441 Table[Length[Select[IntegerPartitions[n],And@@EvenQ/@Rest[Length/@Split[#]]&]],{n,1,30}] (* _Gus Wiseman_, Jan 13 2022 *)
%o A096441 (PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ _Joerg Arndt_, Jan 17 2016
%Y A096441 Bisections are A078408 and A096967.
%Y A096441 The complement in partitions is counted by A006477
%Y A096441 A version for compositions is A016116.
%Y A096441 A pointed version is A035363, ranked by A066207.
%Y A096441 A000041 counts integer partitions.
%Y A096441 A025065 counts palindromic partitions.
%Y A096441 A027187 counts partitions with even length/maximum.
%Y A096441 A035377 counts partitions using multiples of 3.
%Y A096441 A058696 counts partitions of even numbers, ranked by A300061.
%Y A096441 A340785 counts factorizations into even factors.
%Y A096441 Cf. A000009, A002865, A027383, A035457, A117298, A117989, A168021, A274230, A345170, A349060, A349061.
%K A096441 nonn
%O A096441 1,2
%A A096441 Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004

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