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A137966 G.f. satisfies A(x) = 1+x + x^2*A(x)^6.

%I A137966 #15 Aug 23 2023 08:34:06
%S A137966 1,1,1,6,21,86,396,1812,8607,41958,207333,1040234,5281965,27078756,
%T A137966 140021248,729369474,3823598232,20158251814,106809280563,568471343322,
%U A137966 3037782047947,16292380484454,87669285293451,473172657154822
%N A137966 G.f. satisfies A(x) = 1+x + x^2*A(x)^6.
%H A137966 Seiichi Manyama, <a href="/A137966/b137966.txt">Table of n, a(n) for n = 0..1000</a>
%F A137966 a(n) = Sum_{k=0..n-1} C(n-k,k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009
%F A137966 Recurrence: 5*(n-1)*n*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(5*n + 2)*a(n) =  + 576*(n-1)^2*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*a(n-2) + 288*(3*n - 5)*(3*n - 4)*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(11250*n^7 - 111375*n^6 + 440175*n^5 - 888545*n^4 + 975241*n^3 - 574177*n^2 + 165869*n - 18018)*a(n-3) + 144*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 6)*(10125000*n^11 - 218700000*n^10 + 2075585625*n^9 - 11378954250*n^8 + 39836289925*n^7 - 92894908470*n^6 + 145953551806*n^5 - 152681445300*n^4 + 102505633480*n^3 - 41086190160*n^2 + 8557182144*n - 670602240)*a(n-4) + 72*(5*n - 26)*(5*n - 11)*(5*n - 6)*(5*n - 4)*(20250000*n^11 - 569025000*n^10 + 7025658750*n^9 - 50083579125*n^8 + 227686012400*n^7 - 687547140050*n^6 + 1391232445598*n^5 - 1854143517725*n^4 + 1550931293540*n^3 - 737424345140*n^2 + 162058858752*n - 10360465920)*a(n-5) + 144*(5*n - 16)*(5*n - 11)*(5*n - 9)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(202500*n^9 - 5953500*n^8 + 74924775*n^7 - 526434885*n^6 + 2255339082*n^5 - 6025054075*n^4 + 9796892735*n^3 - 8893818500*n^2 + 3545754268*n - 142331280)*a(n-6) + 72*n*(2*n - 9)*(3*n - 17)*(3*n - 10)*(5*n - 21)*(5*n - 16)*(5*n - 14)*(5*n - 11)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(6*n - 41)*(6*n - 13)*a(n-7). - _Vaclav Kotesovec_, Nov 18 2017
%F A137966 a(n) ~ sqrt((1 + 2*r*s^6)/(15*Pi)) / (2*s^2 * n^(3/2) * r^(n + 1/2)), where r = 0.1734895129039028676461340698295316044509963479582... and s = 1.408187415484683441175360883795437925341195617549... are roots of the system of equations 1 + r + r^2*s^6 = s, 6*r^2*s^5 = 1. - _Vaclav Kotesovec_, Nov 18 2017
%t A137966 Flatten[{1, Table[Sum[Binomial[n-k,k]/(n-k) * Binomial[6*k,n-k-1], {k,0,n-1}], {n,1,30}]}] (* _Vaclav Kotesovec_, Nov 18 2017 *)
%o A137966 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^6)^1);polcoeff(A,n)}
%o A137966 (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-k,k)/(n-k)*binomial(6*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009
%Y A137966 Cf. A137967, A137965; A019497, A137954, A137959.
%K A137966 nonn
%O A137966 0,4
%A A137966 _Paul D. Hanna_, Feb 26 2008