cp's OEIS Frontend

A370163 a(0) = 2, a(n) = (-1)^n + (-2)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

%I A370163 #20 Feb 28 2024 20:40:14
%S A370163 2,1,5,25,161,1321,13025,149605,1963841,29004721,475975745,8591917885,
%T A370163 169193833121,3609452038921,82924458549665,2041207822721365,
%U A370163 53594538159184001,1495143168658285921,44164021453758342785,1377005070100813288045,45193800193226286112481
%N A370163 a(0) = 2, a(n) = (-1)^n + (-2)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.
%C A370163 Inverse binomial transform of A370092 + A370456.
%F A370163 E.g.f.: 2*(1 + exp(x))/(1 + exp(x) + exp(2*x) - exp(3*x)).
%o A370163 (SageMath)
%o A370163 def a(m):
%o A370163     if m==0:
%o A370163         return 2
%o A370163     else:
%o A370163         return (-1)^m+(-2)^m+1/2*sum([(1-(-2)^j-(-1)^j)*binomial(m,j)*a(m-j) for j in [1,..,m]])
%o A370163 list(a(m) for m in [0,..,20])
%o A370163 (SageMath)
%o A370163 f=2*(1+e^x)/(1+e^x+e^(2*x)-e^(3*x))
%o A370163 print([(diff(f,x,i)).subs(x=0) for i in [0,..,20]])
%o A370163 (PARI) seq(n)={my(p=exp(x + O(x*x^n))); Vec(serlaplace(2*(1 + p)/(1 + p + p^2 - p^3)))} \\ _Andrew Howroyd_, Feb 26 2024
%Y A370163 Cf. A370092, A370456.
%K A370163 nonn
%O A370163 0,1
%A A370163 _Prabha Sivaramannair_, Feb 26 2024