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 A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices. 7
 1, 2, 2, 4, 3, 7, 5, 11, 8, 17, 12, 26, 18, 37, 27, 54, 38, 76, 54, 106, 76, 145, 104, 199, 142, 266, 192, 357, 256, 472, 340, 621, 448, 809, 585, 1053, 760, 1354, 982, 1740, 1260, 2218, 1610, 2818, 2048, 3559, 2590, 4485, 3264, 5616, 4097, 7018, 5120, 8728, 6378 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of partitions of n such that all differences between successive parts are even, see example. [Joerg Arndt, Dec 27 2012] Number of partitions of n where either all parts are odd or all parts are even. - Omar E. Pol, Aug 16 2013 REFERENCES 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 Karin Baur and Nolan Wallach, Nice parabolic subalgebras of reductive Lie algebras, Represent. Theory 9 (2005), 1-29. A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, arXiv:math-ph/0312030, 2002-2004. FORMULA G.f.: sum(j>=1, q^j * (1-q^j)/prod(i=1..j, 1-q^(2*i) ) ). 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)) ). a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - Vladeta Jovovic, Aug 11 2004 a(n) = A000009(n) + A035363(n) = A000041(n) - A006477(n). - Omar E. Pol, Aug 16 2013 EXAMPLE From Joerg Arndt, Dec 27 2012: (Start) There are a(10)=17 partitions of 10 where all differences between successive parts are even: [ 1]  [ 1 1 1 1 1 1 1 1 1 1 ] [ 2]  [ 2 2 2 2 2 ] [ 3]  [ 3 1 1 1 1 1 1 1 ] [ 4]  [ 3 3 1 1 1 1 ] [ 5]  [ 3 3 3 1 ] [ 6]  [ 4 2 2 2 ] [ 7]  [ 4 4 2 ] [ 8]  [ 5 1 1 1 1 1 ] [ 9]  [ 5 3 1 1 ] [10]  [ 5 5 ] [11]  [ 6 2 2 ] [12]  [ 6 4 ] [13]  [ 7 1 1 1 ] [14]  [ 7 3 ] [15]  [ 8 2 ] [16]  [ 9 1 ] [17]  [ 10 ] (End) MAPLE b:= proc(n, i) option remember; `if`(i>n, 0,       `if`(irem(n, i)=0, 1, 0) +add(`if`(irem(j, 2)=0,        b(n-i*j, i+1), 0), j=0..n/i))     end: a:= n-> b(n, 1): seq(a(n), n=1..60);  # Alois P. Heinz, Mar 26 2014 MATHEMATICA (* The following Mathematica program first generates all of the palindromic, unimodal compositions of n and then counts them. *) 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]] 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 *) PROG (PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ Joerg Arndt, Jan 17 2016 CROSSREFS Bisections are A078408 and A096967. Sequence in context: A265701 A007439 A238622 * A100824 A163227 A325690 Adjacent sequences:  A096438 A096439 A096440 * A096442 A096443 A096444 KEYWORD nonn AUTHOR Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004 STATUS approved

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Last modified April 1 10:22 EDT 2020. Contains 333159 sequences. (Running on oeis4.)