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%I #25 Sep 07 2022 07:28:00
%S 1,1,2,4,9,24,77,299,1419,8312,60452,547939,6213566,88468601,
%T 1585646789,35846274127,1023893974778,37005881297226,1694206791508891,
%U 98335493373334998,7241161595237290969,676871453643079089963,80351261743964014059133,12117563014768206457325416
%N O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - Fibonacci(k)*x).
%C After the first term, row sums of triangle A111669. - _Emanuele Munarini_, Dec 05 2017
%H Alois P. Heinz, <a href="/A135934/b135934.txt">Table of n, a(n) for n = 0..140</a>
%F G.f.: (1 - G(0) )/(1-x) where G(k) = 1 - 1/(1-Fibonacci(k)*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 17 2013
%F G.f.: 1/(x*(1-x)*G(0)) - 1/x where G(k) = 1 - x/(x - 1/(1 + 1/(x*Fibonacci(k)-1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Feb 13 2013
%e A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-x)) + x^3/((1-x)*(1-x)*(1-2*x)) +
%e x^4/((1-x)*(1-x)*(1-2*x)(1-3*x)) + x^5/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)) + x^6/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)*(1-8*x)) +...
%p b:= proc(n, m) option remember; `if`(n=0, 1,
%p (<<0|1>, <1|1>>^m)[1, 2]*b(n-1, m)+b(n-1, m+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 08 2021
%t b[n_, m_] := b[n, m] = If[n == 0, 1,
%t MatrixPower[{{0, 1}, {1, 1}}, m][[1, 2]]*b[n-1, m]+b[n-1, m+1]];
%t a[n_] := b[n, 0];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Sep 07 2022, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-fibonacci(j)*x+x*O(x^n))), n)}
%Y Cf. A000045, A006116, A111669.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 07 2007
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