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A137952
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G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^2.
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6
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1, 1, 2, 7, 24, 95, 386, 1641, 7150, 31844, 144216, 662228, 3076044, 14427582, 68235078, 325049475, 1558212804, 7511319253, 36387218312, 177050945886, 864912345340, 4240388439744, 20857232340528, 102896737106415
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137953.
a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 5*n*(5*n - 4)*(5*n - 3)*(5*n - 1)*(5*n + 3)*(8845200*n^11 - 252428400*n^10 + 3221192232*n^9 - 24137808840*n^8 + 117463352781*n^7 - 387964460127*n^6 + 882822962553*n^5 - 1374856808005*n^4 + 1422227015434*n^3 - 915895407668*n^2 + 320324023880*n - 42693386400)*a(n) = - 360*(5*n - 2)*(5670000*n^13 - 63714600*n^12 - 1032645960*n^11 + 24848001198*n^10 - 218480624507*n^9 + 1101741928166*n^8 - 3582401014336*n^7 + 7865579681092*n^6 - 11836392808433*n^5 + 12130520012664*n^4 - 8236278842764*n^3 + 3497924862840*n^2 - 827741189520*n + 81691545600)*a(n-1) + 180*(2653560000*n^16 - 86342760000*n^15 + 1284348733200*n^14 - 11544882534000*n^13 + 69915022739748*n^12 - 301277354913324*n^11 + 951521048997123*n^10 - 2235356609743737*n^9 + 3921814538564296*n^8 - 5108337175422974*n^7 + 4854490688899951*n^6 - 3250616687965913*n^5 + 1431302003002666*n^4 - 349408874612852*n^3 + 16089460853736*n^2 + 12240998632800*n - 2031289747200)*a(n-2) + 72*(17672709600*n^16 - 601551846000*n^15 + 9383367519936*n^14 - 88661500185240*n^13 + 565349613141438*n^12 - 2565633937621131*n^11 + 8513410651166583*n^10 - 20875837005697545*n^9 + 37705724089968084*n^8 - 49181218885648923*n^7 + 44098626888119141*n^6 - 23771481353637565*n^5 + 3467317211974378*n^4 + 4824415011450004*n^3 - 3654086377331160*n^2 + 1070168332564800*n - 116760296016000)*a(n-3) + 144*(8597534400*n^16 - 305543145600*n^15 + 4975684360704*n^14 - 49077873815616*n^13 + 326509076764188*n^12 - 1543742190898488*n^11 + 5321067950386782*n^10 - 13479709842928188*n^9 + 24903384308348709*n^8 - 32579354322085314*n^7 + 27941366702438094*n^6 - 11913061039189846*n^5 - 3157851308946897*n^4 + 7647346836930652*n^3 - 4534021704525180*n^2 + 1245319349576400*n - 132684717816000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 10)*(6*n - 25)*(6*n - 23)*(8845200*n^11 - 155131200*n^10 + 1183394232*n^9 - 5046898752*n^8 + 12951310413*n^7 - 19922972292*n^6 + 16394061984*n^5 - 2858995378*n^4 - 7011543813*n^3 + 6369403462*n^2 - 2180183136*n + 267092640)*a(n-5). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 4*r*s^3 + 3*r^2*s^6) / (3*Pi*s*(2 + 5*r*s^3))) / (2*n^(3/2) * r^(n + 1/2)), where r = 0.1898739884773465982357897900946346962414966313829... and s = 1.607584028097173055359903977736399386285943742600... are roots of the system of equations 1 + r*(1 + r*s^3)^2 = s, 6*r^2*s^2*(1 + r*s^3) = 1. - Vaclav Kotesovec, Nov 18 2017
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MATHEMATICA
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Flatten[{1, Table[Sum[Binomial[2*(n-k), k]/(n-k) * Binomial[3*k, n-k-1], {k, 0, n-1}], {n, 1, 30}]}] (* Vaclav Kotesovec, Nov 18 2017 *)
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PROG
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(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1+x*(1+x*A^3)^2); polcoeff(A, n)}
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(2*(n-k), k)/(n-k)*binomial(3*k, n-k-1))) \\ Paul D. Hanna, Jun 16 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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