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%I #30 Sep 08 2022 08:46:01
%S 1,4,26,148,867,5048,29428,171512,999653,5826396,33958734,197925996,
%T 1153597255,6723657520,39188347880,228406429744,1331250230601,
%U 7759094953844,45223319492482,263580822001028,1536261612513707,8953988853081192,52187671505973468
%N Expansion of g.f.: exp( Sum_{n>=1} A002203(n)^2 * x^n/n ) where A002203 are the companion Pell numbers.
%H Vincenzo Librandi, <a href="/A204062/b204062.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,10,4,-1).
%F G.f.: 1/((1+x)^2*(1-6*x+x^2)).
%F Self-convolution of A026933.
%F Self-convolution 4th power of A204061.
%F a(n) = Pell(n-1)^2 + a(n-2) where Pell(n) = A000129(n).
%F a(n) = (1/8)*(A001109(n+2) + (-1)^n*(n+2)). - _Bruno Berselli_, Jan 10 2012
%F a(n) = (1/16)*(A000129(2*n+4) + 2*(-1)^n*(n+2)). - _G. C. Greubel_, May 25 2021
%e G.f.: A(x) = 1 + 4*x + 26*x^2 + 148*x^3 + 867*x^4 + 5048*x^5 + ...
%e where
%e log(A(x)) = 2^2*x + 6^2*x^2/2 + 14^2*x^3/3 + 34^2*x^4/4 + 82^2*x^5/5 + 198^2*x^6/6 + 478^2*x^7/7 + ... + A002203(n)^2*x^n/n + ...
%t LinearRecurrence[{4,10,4,-1},{1,4,26,148},30] (* _Vincenzo Librandi_, Feb 12 2012 *)
%t Table[(Fibonacci[2*n+4, 2] + 2*(-1)^n*(n+2))/16, {n, 0, 30}] (* _G. C. Greubel_, May 25 2021 *)
%o (PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
%o {a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2*x^k/k)+x*O(x^n)), n)}
%o (Magma) I:=[1,4,26,148]; [n le 4 select I[n] else 4*Self(n-1) +10*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..31]]; // _G. C. Greubel_, May 25 2021
%o (Sage) [(lucas_number1(2*n+4,2,-1) +2*(-1)^n*(n+2))/16 for n in (0..30)] # _G. C. Greubel_, May 25 2021
%Y Cf. A000129, A002203, A026933, A204061, A212442.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Jan 10 2012
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