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A082733
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Sum of all entries in character table of the symmetric group S_n.
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29
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1, 2, 5, 13, 31, 89, 259, 842, 2810, 10020, 37266, 145373, 586071, 2453927, 10590180, 47159351, 215706629, 1013916313, 4882544468, 24087770591, 121481296510, 626169893024, 3293432146879, 17670096206819, 96589760733604, 537731396393480, 3045955783377644
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OFFSET
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1,2
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LINKS
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Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, On the sum of the entries in a character table, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024), Article #99, 12 pp.
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FORMULA
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Let D(x) = Sum_{n>=0} (2n-1)!!*x^n = 1/(1 - x/(1 - 2x/(1 - 3x/...))) and R_r(x) = Sum_{n>=0} o_r(n)*x^n = 1/(1 - x - r*x^2/(1 - x - 2*r*x^2/(1 - x - 3*r*x^2/...))), where o_r(n) = Sum_{k=0..n/2} binomial(n, 2k)*(2k-1)!!*r^k. Then the generating function of this sequence is Sum_{n>=0} a(n)*x^n = Product_{i >= 1} (D(2ix^{4i}) * R_{2i-1}(x^{2i-1}). - Arvind Ayyer, Jun 11 2024
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EXAMPLE
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a(3) = 5 because the character table of S_3 is / 1 1 1 / 2 0 -1 / 1 -1 1 /.
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MATHEMATICA
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a[n_] := FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"] // Flatten // Total;
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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