cp's OEIS Frontend

From Peter Bala, Dec 22 2021: (Start)

Conjectures:

1) Taking the sequence (a(n))n>=1 modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, ...] with an apparent period of length 6, which divides phi(21) = 12.

2) For i >= 0, define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients.

3) a(m*n) == a(m)^n (mod 2^k) for k = 2*v_2(m) + 4, where v_p(i) denotes the p-adic valuation of i.

4)(i) a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 4

(ii) a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 4. (End)

Reduced numerators of Favard constants.

We say that the sequence S is the exponential limit of the function f relative to the kernel K if and only if the exponential generating functions

egf(n) = Sum_{k=0..n} K(n, k)*f(x*(n-k)) generate a family of sequences

T(n) = k -> (k!/n!)*[x^k] egf(n) which converge to S. Convergence here means that for every fixed k the terms T(n)(k) differ from S(k) only for finitely many indices.

The paradigmatic example is to set f(x) = exp(x), K(n, k) = !k*binomial(n, k) (!n is the subfactorial of n) and obtain for S the Bell numbers. This example is set forth in A320955.

Let D(f)(x) represent the derivative of f(x) with respect to x and (D^(n))(f) the n-th derivative of f. Then the exponential limit of f is B(n)*((D^(n))(f))(0) where B(n) is the n-th Bell number: ExpLim(f) = f(0), (D(f))(0), 2*((D^(2))(f))(0), 5*((D^(3))(f))(0), 15*((D^(4))(f))(0), 52*((D^(5))(f))(0), ... Since exp is a fixed point of D and exp(0) = 1 we have the identity ExpLim(exp)[n] = B(n). Similarly ExpLim(sin)[n] = B(n)*mod(n,2)*(-1)^binomial(n,2).

If we set f = sec + tan and K(n, k) = !k*binomial(n, k) the exponential limit is this sequence, a(n).

DEFINITION

Define polynomials R(n,t) with t = tan(x) by

... (d/dx)^n sec^2(x) = R(n,tan(x))*sec^2(x).

The first few are

... R(0,t) = 1

... R(1,t) = 2*t

... R(2,t) = 2 + 6*t^2

... R(3,t) = 16*t + 24*t^3.

This triangle shows the coefficients of R(n,t) in ascending powers of t called the tangent number triangle in [Hodges and Sukumar].

The polynomials R(n,t) form a companion polynomial sequence to Hoffman's two polynomial sequences - P(n,t) (A155100), the derivative polynomials of the tangent and Q(n,t) (A104035), the derivative polynomials of the secant. See also A008293 and A008294.

COMBINATORIAL INTERPRETATION

A combinatorial interpretation for the polynomial R(n,t) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges].

A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}. They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube.

Let x_1,...,x_n be a signed permutation.

Then 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0.

For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0).

Let sc be the number of sign changes through a snake

... sc = #{i, 1 <= i <= n-1, x_i*x_(i+1) < 0}.

For example, the snake 0 4 -3 -1 -2 0 has sc = 1. The polynomial R(n,t) is the generating function for the sign change statistic on snakes of type S(n+1;0,0):

... R(n,t) = sum {snakes in S(n+1;0,0)} t^sc.

See the example section below for the cases n=1 and n=2.

PRODUCTION MATRIX

Define three arrays R, L, and S as

... R = superdiag[2,3,4,...]

... L = subdiag[1,2,3,...]

... S = diag[2,4,6,...]

with the indicated sequences on the main superdiagonal, the main subdiagonal and main diagonal, respectively, and 0's elsewhere. The array R+L is the production array for this triangle: the first row of (R+L)^n produces the n-th row of the triangle.

On the vector space of complex polynomials the array R, the raising operator, represents the operator p(x) - > d/dx (x^2*p(x)), and the array L, the lowering operator, represents the differential operator d/dx - see Formula (4) below.

The three arrays satisfy the commutation relations

... [R,L] = S, [R,S] = 2*R, [L,S] = -2*L

and hence give a representation of the Lie algebra sl(2).