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Search: id:A096441
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| A096441 |
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Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices. |
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3
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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The Mathematica program actually first generates all of the palindromic, unimodal compositions of n and then counts them.
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REFERENCES
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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. Lie groups and invariant theory, 85-104, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 2005.
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FORMULA
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G.f.: Sum_{j >= 1} q^j (1-q^j)/Product_{1 <= i <= j} (1-q^{2i}).
Another g.f.: F + G - 2, where F = Product_{ j >= 1 } 1/(1-q^2j), G = Product_{ j >= 0 } 1/(1-q^(2j+1)).
a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 11 2004
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MATHEMATICA
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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]]
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CROSSREFS
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Bisections are A078408 and A096967.
Sequence in context: A122585 A057449 A007439 this_sequence A100824 A163227 A048675
Adjacent sequences: A096438 A096439 A096440 this_sequence A096442 A096443 A096444
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KEYWORD
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nonn
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AUTHOR
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Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004
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