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Expansion of e.g.f. 1/(1 + exp(2*x) - exp(3*x)).
4

%I #19 Apr 19 2024 04:35:29

%S 1,1,7,55,571,7471,117307,2148175,44958571,1058555791,27693129307,

%T 796934764495,25018548004171,850870651904911,31163746960955707,

%U 1222922731101304015,51189052318085027371,2276586205163067346831,107204914362429152404507

%N Expansion of e.g.f. 1/(1 + exp(2*x) - exp(3*x)).

%C 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 a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 2^k) * binomial(n,k) * a(n-k).

%F 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 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+exp(2*x)-exp(3*x))))

%o (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 Cf. A371460 (binomial transform).

%Y Cf. A000010, A354242, A355381, A355408, A370092.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jul 01 2022