A278903 - OEIS

%I #16 Apr 14 2022 15:18:10

%S 1,1,20922,96938760190744854628604,

%T 1039473181175725249030299777705981025900981837012416973957739853576960

%N Second series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

%C If we regard Bell(k^2) as the k-th Stieltjes moment for k>=0, then the solution of the Stieltjes moment problem is given in the P. Blasiak et al. reference, see below. We conjecture that a(n)>0 for n>=0. The positivity of these Hankel determinants a(n), n>=0 is one of the conditions for the existence of a positive solution. Apparently this solution is not unique.

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arxiv.org/abs/quant-ph/0303030">Dobinsky-type relations and the log-normal distribution</a>, J. Phys. A: Math. Gen. 36, L273 (2003), arXiv: quant-ph/0303030, 2003.

%p with(LinearAlgebra), with(combinat):

%p h21:=(i, j)->bell((i+j-1)^2):

%p seq(Determinant(Matrix(kk, kk, h21)), kk=0..6);

%t Table[Det[Table[BellB[(i + j - 1)^2], {i, n}, {j, n}]], {n, 5}], n=>1.

%Y Cf. A000110, A277829, A278770, A278868, A278860, A278897.

%K nonn

%O 0,3

%A _Karol A. Penson_, Nov 30 2016