This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355409 #19 Apr 19 2024 04:35:29
%S A355409 1,1,7,55,571,7471,117307,2148175,44958571,1058555791,27693129307,
%T A355409 796934764495,25018548004171,850870651904911,31163746960955707,
%U A355409 1222922731101304015,51189052318085027371,2276586205163067346831,107204914362429152404507
%N A355409 Expansion of e.g.f. 1/(1 + exp(2*x) - exp(3*x)).
%C A355409 Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [1, 1, 7, 1, 4, 1, 1, 1, 7, 1, 4, 1, 1, 1, 7, 1, 4, 1, 1, ...] with an apparent period of 6 = phi(9) beginning at a(1). Cf. A354242. - _Peter Bala_, Apr 16 2024
%F A355409 a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 2^k) * binomial(n,k) * a(n-k).
%F A355409 a(n) ~ n! / ((3 + r^2) * log(r)^(n+1)), where r = (1 + 2*cosh(log((29 + 3*sqrt(93))/2)/3))/3. - _Vaclav Kotesovec_, Jul 01 2022
%o A355409 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+exp(2*x)-exp(3*x))))
%o A355409 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j-2^j)*binomial(i, j)*v[i-j+1])); v;
%Y A355409 Cf. A371460 (binomial transform).
%Y A355409 Cf. A000010, A354242, A355381, A355408, A370092.
%K A355409 nonn
%O A355409 0,3
%A A355409 _Seiichi Manyama_, Jul 01 2022