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O.g.f.: exp( Sum_{n >= 1} A000182(n+1)*x^n/n ), where A000182 is the sequence of tangent numbers.
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%I #12 Sep 27 2015 17:47:47

%S 1,2,10,108,2214,75708,3895236,280356120,26824493574,3287849716332,

%T 501916845156012,93337607623037544,20766799390944491100,

%U 5446109742113077482456,1662395457873577922274888

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

%C 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 First row of square array A262144.

%H P. Bala, <a href="/A100100/a100100.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>

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

%p #A262145

%p #define tangent numbers A000182

%p A000182 := n -> (1/2) * 2^(2*n) * (2^(2*n) - 1) * abs(bernoulli(2*n))/n:

%p a := proc (n) option remember;

%p if n = 0 then 1 else

%p add(A000182(k+1)*a(n-k), k = 1 .. n)/n

%p end if;

%p end proc:

%p seq(a(n), n = 0 .. 15);

%t 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 (Sage)

%o def a_list(n):

%o T = [0]*(n+2); T[1] = 1

%o for k in range(2, n+1): T[k] = (k-1)*T[k-1]

%o for k in range(2, n+1):

%o for j in range(k, n+1): T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]

%o @cached_function

%o def a(n): return sum(T[k+1]*a(n-k) for k in (1..n))//n if n> 0 else 1

%o return [a(k) for k in range(n)]

%o a_list(15) # _Peter Luschny_, Sep 18 2015

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

%K nonn,easy

%O 0,2

%A _Peter Bala_, Sep 13 2015