cp's OEIS Frontend

A373625 Sum of all entries in character table of the hyperoctahedral group B_n.

%I A373625 #18 Oct 18 2024 18:10:04
%S A373625 1,2,8,26,112,410,1860,8074,40376,199050,1085232,5923394,34842408,
%T A373625 206403234,1295653484,8219293954,54613967584,367414298386,
%U A373625 2567777927672,18187100499306,133016727225888,986352813933034,7518613974827732,58110359176236314,460095738657984024
%N A373625 Sum of all entries in character table of the hyperoctahedral group B_n.
%H A373625 Andrew Howroyd, <a href="/A373625/b373625.txt">Table of n, a(n) for n = 0..500</a>
%H A373625 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.
%F A373625 G.f.: Product_{i >= 1} D(4*i*x^(4*i)) * D(2*i*x^(2*i)) * R(2*i-1, x^(2*i-1)), where D(x) is the g.f. of A001147, R(r, x) = Sum_{k>=0} c(r,k)*x^k and c(r,n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * 2^(n-k) * (2*k-1)!! * r^k. [edited by _Andrew Howroyd_, Oct 07 2024]
%F A373625 G.f.: Product_{i >= 1} D(4*i*x^(4*i)) * D(2*i*x^(2*i)) * B(4*i-2, x^(2*i-1)), where D(x) is the g.f. of A001147 and B(k,x) is the g.f. of column k of A376826. - _Andrew Howroyd_, Oct 07 2024
%e A373625 a(2) = 8 because the character table of B_2 is  [[1  1  1  1  1], [ 1 -1 -1  1  1], [ 1 -1  1 -1  1], [ 1  1 -1 -1  1], [ 2  0  0  0 -2]].
%o A373625 (PARI) \\ here B(k,n) is o.g.f. of column k of A376826.
%o A373625 B(k,n)={serlaplace(exp(2*x + k*x^2/2 + O(x*x^n)))}
%o A373625 seq(n)={my(d=serlaplace(1/sqrt(1 - 2*x + O(x*x^(n\2))))); Vec(prod(i=1, (n+1)\2, subst(d + O(x^(n\(2*i)+1)), x, 2*i*x^(2*i))^(2-i%2) * subst(B(4*i-2, n\(2*i-1)), x, x^(2*i-1))))} \\ _Andrew Howroyd_, Oct 07 2024
%Y A373625 Cf. A001147, A376826, A082733.
%K A373625 nonn
%O A373625 0,2
%A A373625 _Arvind Ayyer_, Jun 11 2024
%E A373625 a(0)=1 prepended and a(10) onwards from _Andrew Howroyd_, Oct 06 2024