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%I #9 Mar 25 2014 09:44:48
%S 1,1,2,5,11,26,64,159,402,1032,2677,7010,18510,49220,131691,354282,
%T 957745,2600382,7088008,19388719,53207441,146444424,404151643,
%U 1118132954,3100540971,8615945102,23989662824,66917894562,186983937758,523314016245,1466807316032
%N G.f. satisfies: A(x) = (1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x)).
%H Vaclav Kotesovec, <a href="/A182053/b182053.txt">Table of n, a(n) for n = 0..300</a>
%F Recurrence: 5*(n+2)*(n+3)*(8911*n^7 - 241946*n^6 + 2447725*n^5 - 11084372*n^4 + 19415458*n^3 + 1716316*n^2 - 23882064*n + 2130912)*a(n) = 3*(n+2)*(35644*n^8 - 914318*n^7 + 8350054*n^6 - 29810773*n^5 + 11813540*n^4 + 124863372*n^3 - 96624130*n^2 - 116648601*n + 36899532)*a(n-1) + (17822*n^9 - 448248*n^8 + 3870267*n^7 - 11783352*n^6 - 5744844*n^5 + 67908444*n^4 - 29523545*n^3 - 131900220*n^2 - 65428308*n - 24231312)*a(n-2) + (178220*n^9 - 4749810*n^8 + 45912714*n^7 - 180749091*n^6 + 117079677*n^5 + 903637509*n^4 - 1481741315*n^3 - 502308600*n^2 + 1275968592*n - 5383800)*a(n-3) - 3*(26733*n^9 - 752571*n^8 + 7896778*n^7 - 35859478*n^6 + 44913322*n^5 + 151779908*n^4 - 435106847*n^3 + 135598205*n^2 + 327138534*n - 194418504)*a(n-4) + (17822*n^9 - 528447*n^8 + 6031992*n^7 - 32027385*n^6 + 65161374*n^5 + 61817937*n^4 - 383729654*n^3 + 245358135*n^2 + 217471338*n - 55404648)*a(n-5) + 2*(17822*n^9 - 555180*n^8 + 6785304*n^7 - 40187787*n^6 + 104743872*n^5 + 17193252*n^4 - 685093274*n^3 + 1082675799*n^2 + 285222672*n - 1131441048)*a(n-6) - (17822*n^9 - 581913*n^8 + 7681743*n^7 - 51193035*n^6 + 163962759*n^5 - 87813864*n^4 - 868874108*n^3 + 1698086052*n^2 + 415801800*n - 1243151064)*a(n-7) - (35644*n^9 - 1217292*n^8 + 16363266*n^7 - 106473036*n^6 + 307618491*n^5 - 35551263*n^4 - 1688725327*n^3 + 2463079431*n^2 + 830567142*n - 2522003904)*a(n-8) + (n-8)*(35644*n^8 - 985606*n^7 + 10067008*n^6 - 43505557*n^5 + 43567930*n^4 + 201854264*n^3 - 417755448*n^2 - 116443773*n + 335593314)*a(n-9) - (n-9)*(n-8)*(8911*n^7 - 179569*n^6 + 1183180*n^5 - 2163052*n^4 - 4971815*n^3 + 14491649*n^2 + 4308780*n - 9489060)*a(n-10). - _Vaclav Kotesovec_, Mar 25 2014
%F a(n) ~ sqrt((s*(2-r-r^3*(s-1)+r^5*s))/(1+r+r^2+3*r^3*s))/ (2*sqrt(Pi)* n^(3/2)*r^(n+3/2)), where r = 0.34048516736982998257..., s = 3.7980384578075501949... are roots of the system of equations r + r^2 + r^3 + 2*r^3*s + 2*r^4*s + 2*r^5*s + 3*r^6*s^2 = 1, and (1 + r*s)*(1 + r^2*s)*(1 + r^3*s) = s. - _Vaclav Kotesovec_, Mar 25 2014
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 64*x^6 + 159*x^7 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 36*x^4 + 94*x^5 + 249*x^6 + 660*x^7 +...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 81*x^4 + 231*x^5 + 656*x^6 + 1848*x^7 +...
%e where A(x) = 1 + x*(1+x+x^2)*A(x) + x^3*(1+x+x^2)*A(x)^2 + x^6*A(x)^3.
%e The logarithm of the g.f. begins:
%e log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 168*x^6/6 + 456*x^7/7 + 1255*x^8/8 + 3493*x^9/9 + 9753*x^10/10 +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x^2*A)*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A004148.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 08 2012
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