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

0,2

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

Table of n, a(n) for n=0..14.

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

Formula

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A355210 A185396 A003222 * A355209 A343307 A003167

Adjacent sequences: A262142 A262143 A262144 * A262146 A262147 A262148

Keyword

nonn,easy

Author

Peter Bala, Sep 13 2015

Status

approved