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%I #53 Nov 28 2023 00:20:05
%S 1,1,2,8,44,316,2776,28912,347888,4750064,72548576,1225540096,
%T 22686824512,456700011328,9932944782464,232113573798656,
%U 5799735585095936,154302762658308352,4354977806995644928,129961223706359609344,4088626884119702752256,135246429574930409348096
%N E.g.f.: exp( Sum_{n>=1} Fibonacci(n)*x^n/n ).
%H Vaclav Kotesovec, <a href="/A244430/b244430.txt">Table of n, a(n) for n = 0..410</a>
%H Tomislav Doslic and R. Sharafdini, <a href="https://doi.org/10.1007/978-3-319-31584-3_10">Hosoya Index of Splices, Bridges, and Necklaces</a>, in Distance, Symmetry, and Topology in Carbon Nanomaterials, 2016, pp 147-156. Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 9), doi:10.1007/978-3-319-31584-3_10. See page 9, at the end of section 2, where a sequence with this recurrence although with different initial conditions is mentioned.
%F E.g.f.: ( (1 + x/Phi) / (1 - Phi*x) )^(sqrt(5)/5) where Phi = (sqrt(5)+1)/2.
%F E.g.f.: exp( Integral 1/(1-x-x^2) dx ).
%F a(n) ~ n! * 5^(1/(2*sqrt(5))) * n^(1/sqrt(5)-1) * ((1+sqrt(5))/2)^(n-1/sqrt(5)) / GAMMA(1/sqrt(5)). - _Vaclav Kotesovec_, Jun 28 2014
%F a(n) = n*a(n-1) + (n-1)*(n-2)*a(n-2) for a(0)=a(1)=1. - _G. C. Greubel_, May 02 2015
%F E.g.f.: sqrt(5)*Phi^(-2/sqrt(5))*( B((Phi + x)/sqrt(5); 2/(sqrt(5) Phi), (2 Phi)/sqrt(5)) - B(Phi/sqrt(5); 2/(sqrt(5) Phi), (2 Phi)/sqrt(5)) ), where B(z; a, b) is the Incomplete Beta function and a(0)=0, a(1)=a(2)=1. - _G. C. Greubel_, May 02 2015
%F 0 = a(n)*(-72*a(n+2) +384*a(n+3) +772*a(n+4) -12*a(n+5) -64*a(n+6) +5*a(n+7)) +a(n+1)*(-124*a(n+3) -234*a(n+4) +74*a(n+5)) +a(n+2)*(+12*a(n+2) -54*a(n+3) -20*a(n+4) -30*a(n+5)) +a(n+3)*(+10*a(n+3) +25*a(n+4)) for all n>=0. - _Michael Somos_, Sep 24 2022
%F E.g.f.: exp((2/sqrt(5)) * arctanh(x*sqrt(5)/(2-x))). - _Fabian Pereyra_, Oct 11 2023
%F a(n) = n!*Sum_{k=0..n} binomial(n-1,k-1)*binomial(1/sqrt(5),k)*((1+sqrt(5))/2)^(n-k)*(sqrt(5))^k. - _Fabian Pereyra_, Oct 19 2023
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 44*x^4/4! + 316*x^5/5! + 2776*x^6/6! +...
%e where
%e log(A(x)) = x + x^2/2 + 2*x^3/3 + 3*x^4/4 + 5*x^5/5 + 8*x^6/6 + 13*x^7/7 + 21*x^8/8 + 34*x^9/9 +...+ A000045(n)*x^n/n +...
%p f:= gfun:-rectoproc({a(n) = n*a(n-1) + (n-1)*(n-2)*a(n-2),a(0)=1,a(1)=1},a(n),remember):
%p seq(f(n), n=0..50); # _Robert Israel_, May 22 2015
%t a[ n_ ] := a[n] =(n-1)*(n-2)a[n-2]+n*a[n-1]; a[0] := 1; a[1] := 1 (* _G. C. Greubel_, May 02 2015 *)
%t RecurrenceTable[{a[0]==1, a[1]==1, a[n]==(n-1)*(n-2)a[n-2]+n*a[n-1]}, a, {n, 20}] (* _G. C. Greubel_, May 02 2015 *)
%o (PARI) {a(n)=n!*polcoeff(exp(intformal(1/(1-x-x^2 +x*O(x^n)))),n)}
%o for(n=0,30,print1(a(n),", "))
%o (Magma) [n le 2 select 1 else (n-1)*Self(n-1)+(n^2-5*n+6)*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, May 03 2015
%Y Cf. A244451, A244432, A000045.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 27 2014
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