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A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).

%I A320352 #16 Aug 20 2025 00:30:17
%S A320352 0,1,3,19,159,1651,20583,299419,4977759,93097891,1934655063,
%T A320352 44224195819,1102820674959,29792843865331,866769668577543,
%U A320352 27018340680076219,898343366411181759,31736205208791131971,1187110673532381604023,46871464129796857140619,1948059531745350527058159
%N A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).
%C A320352 From _Peter Bala_, Aug 19 2025: (Start)
%C A320352 Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, ...] with an apparent period of 6 = phi(9) beginning at n = 2. Cf. A004123.(End)
%H A320352 Seiichi Manyama, <a href="/A320352/b320352.txt">Table of n, a(n) for n = 0..401</a>
%F A320352 E.g.f.: (1 + sinh(x) - cosh(x))/(1 - 2*sinh(x)).
%F A320352 a(n) = Sum_{k=0..n} Stirling2(n,k)*Fibonacci(k)*k!.
%F A320352 a(n) ~ n! / (sqrt(5) * phi^2 * (log(phi))^(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 12 2018
%p A320352 seq(n!*coeff(series((exp(x) - 1)/(exp(x) - exp(2*x) + 1), x=0, 22), x, n), n=0..21); # _Paolo P. Lava_, Jan 09 2019
%t A320352 nmax = 20; CoefficientList[Series[(Exp[x] - 1)/(Exp[x] - Exp[2 x] + 1), {x, 0, nmax}], x] Range[0, nmax]!
%t A320352 Table[Sum[StirlingS2[n, k] Fibonacci[k] k!, {k, 0, n}], {n, 0, 20}]
%Y A320352 Cf. A000045, A000556, A000557, A005443, A005445, A048993, A263576.
%K A320352 nonn
%O A320352 0,3
%A A320352 _Ilya Gutkovskiy_, Oct 11 2018