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%I #29 Feb 02 2021 08:43:41
%S 1,3,17,99,609,3843,24689,160611,1054657,6975747,46406097,310171491,
%T 2081258529,14011445763,94594402353,640188979299,4341874207617,
%U 29502747778563,200803974858641,1368767759442531,9342637825548769,63846282803069187,436797192815981553,2991302112253485411,20504081077963103041,140665546932766467843,965770879590646638929,6635507385062085656931,45621050527781298148257
%N G.f.: 1/sqrt(1 - 6*x - 7*x^2).
%H Robert Israel, <a href="/A322242/b322242.txt">Table of n, a(n) for n = 0..1184</a>
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k).
%F a(n) = Sum_{k=0..n} 7^(n-k) * (-2)^k * binomial(n,k)*binomial(2*k,k).
%F a(n) equals the (central) coefficient of x^n in (1 + 3*x + 4*x^2)^n.
%F exp( Sum_{n>=1} a(n)*x^n/n ) = (1-3*x - sqrt(1 - 6*x - 7*x^2))/(8*x^2).
%F D-finite with recurrence: (7*n+7)*a(n) + (9+6*n)*a(n+1) + (-n-2)*a(n+2) = 0. - _Robert Israel_, Dec 10 2018
%F a(n)^2 = A322243(n), which gives the coefficients in 1 / AGM(1+7*x, sqrt((1-x)*(1-7^2*x))). - _Paul D. Hanna_, Apr 20 2019
%F a(n) ~ 7^(n + 1/2) / (2^(3/2)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Sep 27 2019
%F E.g.f.: exp(3*x) * BesselI(0,4*x). - _Ilya Gutkovskiy_, Feb 02 2021
%F a(n) = 7^n*2F1([1/2, -n], [1], 8/7), where 2F1 is the hypergeometric function. - _Stefano Spezia_, Feb 02 2021
%e G.f.: A(x) = 1 + 3*x + 17*x^2 + 99*x^3 + 609*x^4 + 3843*x^5 + 24689*x^6 + 160611*x^7 + 1054657*x^8 + 6975747*x^9 + 46406097*x^10 + ...
%e such that A(x)^2 = 1/(1 - 6*x - 7*x^2).
%e RELATED SERIES.
%e Ignoring the initial term, this sequence yields the logarithmic derivative of
%e exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 3*x + 13*x^2 + 63*x^3 + 329*x^4 + 1803*x^5 + 10229*x^6 + 59559*x^7 + 353873*x^8 + 2136915*x^9 + 13076637*x^10 + ...
%e which equals (1-3*x - sqrt(1 - 6*x - 7*x^2))/(8*x^2).
%p f:= gfun:-rectoproc({(7*n+7)*a(n)+(9+6*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 3}, a(n), remember):
%p map(f, [$0..30]); # _Robert Israel_, Dec 10 2018
%t CoefficientList[Series[1/Sqrt[1-6x-7x^2],{x,0,40}],x] (* _Harvey P. Dale_, Apr 14 2019 *)
%o (PARI) /* Using generating function: */
%o {a(n) = polcoeff( 1/sqrt(1 - 6*x - 7*x^2 +x*O(x^n)),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* Using binomial formula: */
%o {a(n) = sum(k=0,n, (-1)^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* Using binomial formula: */
%o {a(n) = sum(k=0,n, 7^(n-k)*(-2)^k*binomial(n,k)*binomial(2*k,k))}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* a(n) is central coefficient in (1 + 3*x + 4*x^2)^n */
%o {a(n) = polcoeff( (1 + 3*x + 4*x^2 +x*O(x^n))^n, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A322243.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 08 2018
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