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A143560 G.f. satisfies: A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)^2/A(-x)^2. 1

%I

%S 1,1,3,6,16,38,110,276,818,2158,6528,17766,54622,151852,472674,

%T 1334886,4195328,11992486,37981982,109622228,349384626,1016304750,

%U 3256170672,9533400198,30680043630,90318157804,291763419458,862944630022

%N G.f. satisfies: A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)^2/A(-x)^2.

%H Vaclav Kotesovec, <a href="/A143560/b143560.txt">Table of n, a(n) for n = 0..360</a>

%F Recurrence: (n-1)*(n+2)*(5831*n^8 - 243383*n^7 + 4206188*n^6 - 38710080*n^5 + 202276387*n^4 - 584893435*n^3 + 802855108*n^2 - 210580008*n - 347616720)*a(n) = 3*(12838*n^8 - 601923*n^7 + 11429743*n^6 - 112875831*n^5 + 613634611*n^4 - 1744584366*n^3 + 1944017772*n^2 + 916460124*n - 2513261160)*a(n-1) + (58310*n^10 - 2550450*n^9 + 46725161*n^8 - 462958323*n^7 + 2657843883*n^6 - 8635850811*n^5 + 13133286592*n^4 + 1880730912*n^3 - 32815374482*n^2 + 34499731776*n - 8930250360)*a(n-2) + 3*(158319*n^8 - 5967654*n^7 + 91632286*n^6 - 732085326*n^5 + 3196674685*n^4 - 7121402136*n^3 + 5458082266*n^2 + 4616459592*n - 6817957680)*a(n-3) + (29155*n^10 - 1362690*n^9 + 27450577*n^8 - 310424661*n^7 + 2141375514*n^6 - 9068431947*n^5 + 21700482026*n^4 - 18331373994*n^3 - 37933350100*n^2 + 108779883744*n - 81041494680)*a(n-4) + 3*(88151*n^8 - 3480764*n^7 + 58566564*n^6 - 549211462*n^5 + 3139914339*n^4 - 11150789950*n^3 + 23564857474*n^2 - 25545504056*n + 8635994760)*a(n-5) + 3*(34986*n^10 - 1740186*n^9 + 37087582*n^8 - 440043727*n^7 + 3147927467*n^6 - 13589282015*n^5 + 32153246357*n^4 - 23663011820*n^3 - 62023415792*n^2 + 145867948748*n - 88292684640)*a(n-6) + 9*(n-9)*(2205*n^7 - 64442*n^6 + 1342883*n^5 - 21623864*n^4 + 202409696*n^3 - 984690494*n^2 + 2243960384*n - 1759198080)*a(n-7) + 27*(n-10)*(n-9)*(5831*n^8 - 196735*n^7 + 2665775*n^6 - 18257459*n^5 + 63708572*n^4 - 86956796*n^3 - 67122630*n^2 + 279585146*n - 172700112)*a(n-8). - _Vaclav Kotesovec_, Mar 26 2014

%F a(n) ~ c / (sqrt(Pi)*n^(3/2)*r^n), where r = sqrt((2*sqrt(6*(sqrt(21)-1))-3-sqrt(21))/2)/3 = 0.306418592573502104049... and c = sqrt(p) + sqrt(q)/6 = 3.2890504530125029005631376... if n is even, c = sqrt(p) - sqrt(q)/6 = 3.165028435691778505455199... if n is odd, where p = 10.413783575404573013... is the root of the equation 1806589575 - 477356392800*p + 2025053442720*p^2 - 16252127136000*p^3 + 272533795072*p^4 + 5239763472384*p^5 + 4193987715072*p^6 - 2988802768896*p^7 + 243799621632*p^8 = 0, and q = 0.1384331470227185568... is the root of the equation 11853034201575 - 86998202587800*q + 10251833053770*q^2 - 2285455378500*q^3 + 1064585137*q^4 + 568550724*q^5 + 12641022*q^6 - 250236*q^7 + 567*q^8 = 0. - _Vaclav Kotesovec_, Mar 26 2014

%e A(x) = 1 + x + 3*x^2 + 6*x^3 + 16*x^4 + 38*x^5 + 110*x^6 + 276*x^7 +...

%e A(x)/A(-x) = 1 + 2*x + 2*x^2 + 8*x^3 + 14*x^4 + 46*x^5 + 96*x^6 +...

%e A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 24*x^3 + 64*x^4 + 180*x^5 +...

%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,B=A/subst(A,x,-x);A=1+x*B+x^2*B^2);polcoeff(A,n)}

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 24 2008

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Last modified August 10 08:49 EDT 2022. Contains 356039 sequences. (Running on oeis4.)