A082733 - OEIS

%I #52 Oct 13 2024 00:17:40

%S 1,2,5,13,31,89,259,842,2810,10020,37266,145373,586071,2453927,

%T 10590180,47159351,215706629,1013916313,4882544468,24087770591,

%U 121481296510,626169893024,3293432146879,17670096206819,96589760733604,537731396393480,3045955783377644

%N Sum of all entries in character table of the symmetric group S_n.

%H Ludovic Schwob, <a href="/A082733/b082733.txt">Table of n, a(n) for n = 1..200</a>

%H Ron M. Adin, Alexander Postnikov, and Yuval Roichman! <a href="https://doi.org/10.1016/j.jalgebra.2008.03.030">Combinatorial Gelfand models</a>, J. Algebra, 320(3):1311-1325, 2008.

%H Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, <a href="https://arxiv.org/abs/2406.06036">How large is the character degree sum compared to the character table sum for a finite group?</a>, arXiv preprint arXiv:2406.06036 [math.RT], 2024. See p. 5.

%H Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2024/99.pdf">On the sum of the entries in a character table</a>, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024), Article #99, 12 pp.

%H J. B. Geloun and S. Ramgoolam, <a href="http://arxiv.org/abs/1307.6490">Counting Tensor Model Observables and Branched Covers of the 2-Sphere</a>, arXiv preprint arXiv:1307.6490 [hep-th], 2013.

%F 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

%e a(3) = 5 because the character table of S_3 is / 1 1 1 / 2 0 -1 / 1 -1 1 /.

%t a[n_] := FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"] // Flatten // Total;

%t Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 10}] (* _Jean-François Alcover_, Nov 03 2018 *)

%o (GAP) A082733 := n -> Sum(Sum(Irr(CharacterTable("Symmetric", n)))); # _Eric M. Schmidt_, Jul 03 2012

%Y Cf. A086808, A085624, A085646.

%K nonn

%O 1,2

%A _Vladeta Jovovic_, May 20 2003

%E More terms from _Eric M. Schmidt_, Jul 03 2012