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
Programs
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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);
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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).
Comments