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A371984 Binomial transform of A371460.

%I A371984 #10 May 11 2024 16:51:54
%S A371984 1,3,15,117,1227,16053,251955,4613997,96566667,2273672133,59482039395,
%T A371984 1711735382877,53737315411707,1827584253650613,66936582030410835,
%U A371984 2626714554845111757,109948916113808074347,4889877314768678051493
%N A371984 Binomial transform of A371460.
%F A371984 a(0) = 1, a(n) = Sum_{j=1..n} (1-(-2)^j)*binomial(n,j)*a(n-j) for n > 0.
%F A371984 a(0) = 1, a(n) = 2^n + Sum_{j=1..n} (3^j-2^j)*binomial(n,j)*a(n-j) for n > 0.
%F A371984 E.g.f.: exp(2*x)/(1 + exp(2*x) - exp(3*x)).
%t A371984 nn = 17; a[0] = 1; Do[Set[a[n], 2^n + Sum[(3^j - 2^j)*Binomial[n, j]*a[n - j], {j, n}]], {n, nn}]; Array[a, nn + 1, 0] (* _Michael De Vlieger_, Apr 19 2024 *)
%o A371984 (SageMath)
%o A371984 def a(n):
%o A371984     if n==0:
%o A371984         return 1
%o A371984     else:
%o A371984         return sum([(1-(-2)^j)*binomial(n,j)*a(n-j) for j in [1,..,n]])
%o A371984 list(a(n) for n in [0,..,20])
%o A371984 (SageMath)
%o A371984 f= e^(2*x)/(1 + e^(2*x) - e^(3*x))
%o A371984 print([(diff(f,x,i)).subs(x=0) for i in [0,..,20]])
%Y A371984 Cf. A371460.
%K A371984 nonn
%O A371984 0,2
%A A371984 _Prabha Sivaramannair_, Apr 15 2024