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

1, 2, 8, 26, 112, 410, 1860, 8074, 40376, 199050, 1085232, 5923394, 34842408, 206403234, 1295653484, 8219293954, 54613967584, 367414298386, 2567777927672, 18187100499306, 133016727225888, 986352813933034, 7518613974827732, 58110359176236314, 460095738657984024

Offset: 0

Author

Arvind Ayyer, Jun 11 2024

Keywords

• nonn

Examples

			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]].
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036, [math.RT], 2024.

Crossrefs

Cf. A001147, A376826, A082733.

Programs

PARI

\\ here B(k,n) is o.g.f. of column k of A376826.
B(k,n)={serlaplace(exp(2*x + k*x^2/2 + O(x*x^n)))}
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

Formula

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]

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

Extensions

a(0)=1 prepended and a(10) onwards from Andrew Howroyd, Oct 06 2024