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A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

%I A096054 #22 May 08 2025 02:23:16
%S A096054 1,91,3751,138811,5028751,181308931,6529545751,235085301451,
%T A096054 8463265086751,304679288612371,10968470088963751,394865064451017691,
%U A096054 14215143591303768751,511745180725868773411,18422826609078989373751,663221758853362301815531,23875983327059668074930751
%N A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.
%H A096054 Colin Barker, <a href="/A096054/b096054.txt">Table of n, a(n) for n = 1..600</a>
%H A096054 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (50,-553,1800,-1296).
%F A096054 a(n) = (1/12)*(36^n - 2*9^n - 3*4^n+6).
%F A096054 From _Colin Barker_, May 30 2020: (Start)
%F A096054 G.f.: x*(1 - 6*x)*(1 + 47*x + 36*x^2) / ((1 - x)*(1 - 4*x)*(1 - 9*x)*(1 - 36*x)).
%F A096054 a(n) = 50*a(n-1) - 553*a(n-2) + 1800*a(n-3) - 1296*a(n-4) for n>4. (End)
%t A096054 a[n_] := 6^(2*n-1) * BernoulliB[2*n, 1/6] / BernoulliB[2*n]; Array[a, 15] (* _Amiram Eldar_, May 07 2025 *)
%o A096054 (PARI) a(n)=(1/12)*36^n-(1/6)*9^n-(1/4)*4^n+1/2;
%Y A096054 Cf. A083420, A096045, A096046, A096047, A096048, A096053.
%K A096054 nonn,easy
%O A096054 1,2
%A A096054 _Benoit Cloitre_, Jun 18 2004