A278868 -id:A278868 - cp's OEIS frontend

A278897 First series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

1, 1, 14, 146275425484, 558429168112511379835233509679413804180016

Offset: 0

Karol A. Penson, Nov 30 2016

nonn

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.

P. Blasiak, K. A. Penson and A. I. Solomon, Dobinsky-type relations and the log-normal distribution, J. Phys. A: Math. Gen. 36, L273 (2003), arXiv: quant-ph/0303030, 2003.

Cf. A000110, A277829, A278770, A278868, A278860.

with(LinearAlgebra), with(combinat):
h20:=(i,j)->bell((i+j-2)^2):
seq(Determinant(Matrix(kk,kk,h20)),kk=0..6);

Table[Det[Table[BellB[(i + j - 2)^2], {i, n}, {j, n}]], {n, 6}], n=>1.

A278903 Second series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

Original entry on oeis.org

1, 1, 20922, 96938760190744854628604, 1039473181175725249030299777705981025900981837012416973957739853576960

Offset: 0

Karol A. Penson, Nov 30 2016

nonn

P. Blasiak, K. A. Penson and A. I. Solomon, Dobinsky-type relations and the log-normal distribution, J. Phys. A: Math. Gen. 36, L273 (2003), arXiv: quant-ph/0303030, 2003.

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

with(LinearAlgebra), with(combinat):
h21:=(i, j)->bell((i+j-1)^2):
seq(Determinant(Matrix(kk, kk, h21)), kk=0..6);

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

cp's OEIS Frontend

A278897 First series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

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