A262145 O.g.f.: exp( Sum_{n >= 1} A000182(n+1)*x^n/n ), where A000182 is the sequence of tangent numbers.

1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, 3287849716332, 501916845156012, 93337607623037544, 20766799390944491100, 5446109742113077482456, 1662395457873577922274888

Offset: 0

Author

Peter Bala, Sep 13 2015

Keywords

• nonn
• easy

Comments

It appears that the sequence has integer entries. Calculation suggests the following conjecture: the expansion of exp( Sum_{n >= 1} A000182(n + m)*x^n/n ) has integer coefficients for m = 1, 2, 3, .... This is the case m = 1. Cf. A255881 and A255895.

First row of square array A262144.

Links

• P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Crossrefs

Cf. A000182, A255881, A255895, A262144 (first row).

Programs

Maple

#A262145
#define tangent numbers A000182
A000182 := n -> (1/2) * 2^(2*n) * (2^(2*n) - 1) * abs(bernoulli(2*n))/n:
a := proc (n) option remember;
if n = 0 then 1 else
  add(A000182(k+1)*a(n-k), k = 1 .. n)/n
end if;
end proc:
seq(a(n), n = 0 .. 15);

Mathematica

max = 15; CoefficientList[E^Sum[(-1)^n*2^(2*n+1)*(4^(n+1)-1)*BernoulliB[2*(n+1)]*x^n / (n*(n+1)), {n, 1, max}] + O[x]^max, x] (* Jean-François Alcover, Sep 18 2015 *)

Sage

def a_list(n):
    T = [0]*(n+2); T[1] = 1
    for k in range(2, n+1): T[k] = (k-1)*T[k-1]
    for k in range(2, n+1):
        for j in range(k, n+1): T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
    @cached_function
    def a(n): return sum(T[k+1]*a(n-k) for k in (1..n))//n if n> 0 else 1
    return [a(k) for k in range(n)]
a_list(15) # Peter Luschny, Sep 18 2015

Formula

Recurrence: a(n) = 1/n * Sum_{k = 1..n} A000182(k+1)*a(n-k).