A137953 - OEIS

%I #15 Mar 03 2018 13:53:25

%S 1,1,3,9,34,132,546,2327,10191,45534,206788,951723,4429182,20808186,

%T 98550468,470038119,2255684699,10883852112,52769785320,256960840946,

%U 1256147650818,6162349332204,30328107189312,149698391878458

%N G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^3.

%H Vincenzo Librandi, <a href="/A137953/b137953.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137952.

%F a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009

%F Recurrence: 5*n*(5*n-3)*(5*n-2)*(5*n+1)*(5*n+4)*(2948400*n^11 - 80922240*n^10 + 991552680*n^9 - 7191167904*n^8 + 34388915791*n^7 - 113938412552*n^6 + 266574560812*n^5 - 439214051186*n^4 + 497527715029*n^3 - 367402366838*n^2 + 158427508008*n - 30063700800)*a(n) = -240*(5*n-1)*(3402000*n^13 - 102564900*n^12 + 1682146080*n^11 - 16176231033*n^10 + 95359496344*n^9 - 359981654612*n^8 + 893831335718*n^7 - 1468770570635*n^6 + 1566970769558*n^5 - 1019176919948*n^4 + 331927521052*n^3 + 34505928*n^2 - 32180612832*n + 6541274880)*a(n-1) + 180*(884520000*n^16 - 26930232000*n^15 + 372745486800*n^14 - 3118060887120*n^13 + 17644263763548*n^12 - 71507400823524*n^11 + 214013670957835*n^10 - 480132169105811*n^9 + 810380315383846*n^8 - 1022562903644722*n^7 + 947982058983979*n^6 - 624324084479227*n^5 + 273663045967416*n^4 - 68343334466444*n^3 + 4273926176256*n^2 + 2065304121408*n - 381518968320)*a(n-2) + 72*(5890903200*n^16 - 188191699920*n^15 + 2743292998800*n^14 - 24248455085592*n^13 + 145518104758338*n^12 - 628264374415281*n^11 + 2014705595228766*n^10 - 4876859081303636*n^9 + 8950855221646414*n^8 - 12378944029917433*n^7 + 12665670452628658*n^6 - 9249292270917382*n^5 + 4496305419163048*n^4 - 1229711760456116*n^3 + 68797455703176*n^2 + 53468550934560*n - 10544040864000)*a(n-3) + 72*(5731689600*n^16 - 191702972160*n^15 + 2927459413440*n^14 - 27105381081216*n^13 + 170350803352728*n^12 - 770345146059408*n^11 + 2589617705669352*n^10 - 6581794624393248*n^9 + 12710327685293639*n^8 - 18531898603387194*n^7 + 20012311600272546*n^6 - 15421584075698196*n^5 + 7904537517669183*n^4 - 2290793383663938*n^3 + 159318295564312*n^2 + 94065554487360*n - 19593691084800)*a(n-4) + 72*(2*n-9)*(3*n-11)*(3*n-7)*(6*n-25)*(6*n-23)*(2948400*n^11 - 48489840*n^10 + 344492280*n^9 - 1422208584*n^8 + 3817772239*n^7 - 6909787807*n^6 + 8311308487*n^5 - 6272196721*n^4 + 2621759746*n^3 - 403021048*n^2 - 67705152*n + 22579200)*a(n-5). - _Vaclav Kotesovec_, Mar 25 2014

%F a(n) ~ sqrt(3*s*(s-1)*(3*s-2)/(5*s-3)) / (2*sqrt(Pi)*n^(3/2)*r^n), where s = 1.7888356349988794022183... is the root of the equation 216*(s-1)^2 = s*(5*s-6)^4, and r = 1/(s*(5*s-6)) = 0.189873988477346598... - _Vaclav Kotesovec_, Mar 25 2014

%t Flatten[{1,Table[Sum[Binomial[3*(n-k), k]/(n-k)*Binomial[2*k, n-k-1],{k,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_ after _Paul D. Hanna_, Mar 25 2014 *)

%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^3);polcoeff(A,n)}

%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(n-k),k)/(n-k)*binomial(2*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009

%Y Cf. A137952, A137954; A137957, A137962, A137969.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 26 2008