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A262145 O.g.f.: exp( Sum_{n >= 1} A000182(n+1)*x^n/n ), where A000182 is the sequence of tangent numbers. 3
1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, 3287849716332, 501916845156012, 93337607623037544, 20766799390944491100, 5446109742113077482456, 1662395457873577922274888 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A306064 A185396 A003222 * A003167 A240625 A062412

Adjacent sequences:  A262142 A262143 A262144 * A262146 A262147 A262148

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Sep 13 2015

STATUS

approved

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Last modified January 25 19:09 EST 2020. Contains 331249 sequences. (Running on oeis4.)