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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.

%I A262145 #12 Sep 27 2015 17:47:47
%S A262145 1,2,10,108,2214,75708,3895236,280356120,26824493574,3287849716332,
%T A262145 501916845156012,93337607623037544,20766799390944491100,
%U A262145 5446109742113077482456,1662395457873577922274888
%N A262145 O.g.f.: exp( Sum_{n >= 1} A000182(n+1)*x^n/n ), where A000182 is the sequence of tangent numbers.
%C A262145 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.
%C A262145 First row of square array A262144.
%H A262145 P. Bala, <a href="/A100100/a100100.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>
%F A262145 Recurrence: a(n) = 1/n * Sum_{k = 1..n} A000182(k+1)*a(n-k).
%p A262145 #A262145
%p A262145 #define tangent numbers A000182
%p A262145 A000182 := n -> (1/2) * 2^(2*n) * (2^(2*n) - 1) * abs(bernoulli(2*n))/n:
%p A262145 a := proc (n) option remember;
%p A262145 if n = 0 then 1 else
%p A262145   add(A000182(k+1)*a(n-k), k = 1 .. n)/n
%p A262145 end if;
%p A262145 end proc:
%p A262145 seq(a(n), n = 0 .. 15);
%t A262145 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 *)
%o A262145 (Sage)
%o A262145 def a_list(n):
%o A262145     T = [0]*(n+2); T[1] = 1
%o A262145     for k in range(2, n+1): T[k] = (k-1)*T[k-1]
%o A262145     for k in range(2, n+1):
%o A262145         for j in range(k, n+1): T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
%o A262145     @cached_function
%o A262145     def a(n): return sum(T[k+1]*a(n-k) for k in (1..n))//n if n> 0 else 1
%o A262145     return [a(k) for k in range(n)]
%o A262145 a_list(15) # _Peter Luschny_, Sep 18 2015
%Y A262145 Cf. A000182, A255881, A255895, A262144 (first row).
%K A262145 nonn,easy
%O A262145 0,2
%A A262145 _Peter Bala_, Sep 13 2015