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Row 6 of table A111528.
11

%I #36 Jan 21 2020 10:26:19

%S 1,1,8,78,876,10956,149472,2195208,34398288,571525200,10022997888,

%T 184897670112,3578224662720,72486450479808,1534267158087168,

%U 33877135427154048,779208751651730688,18645519786163266816

%N Row 6 of table A111528.

%H Vaclav Kotesovec, <a href="/A111533/b111533.txt">Table of n, a(n) for n = 0..440</a>

%H A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

%F G.f.: (1/6)*log(Sum_{n>=0} (n+5)!/5!*x^n) = Sum_{n>=1} a(n)*x^n/n.

%F G.f.: 1/(1 + 6*x - 7*x/(1 + 7*x - 8*x/(1 + 8*x -... (continued fraction).

%F a(n) = Sum_{k=0..n} 6^(n-k)*A089949(n,k). - _Philippe Deléham_, Oct 16 2006

%F G.f.: (5 + 1/Q(0))/6, where Q(k) = 1 - 4*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 04 2013

%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 05 2013

%F a(n) ~ n! * n^6/6! * (1 + 9/n + 19/n^2 - 69/n^3 - 704/n^4 - 5880/n^5 - 65736/n^6 - 896832/n^7 - 14068080/n^8 - 246800304/n^9 - 4760585136/n^10). - _Vaclav Kotesovec_, Jul 27 2015

%F From _Peter Bala_, May 25 2017: (Start)

%F O.g.f.: A(x) = ( Sum_{n >= 0} (n+6)!/6!*x^n ) / ( Sum_{n >= 0} (n+5)!/5!*x^n ).

%F 1/(1 - 6*x*A(x)) = Sum_{n >= 0} (n+5)!/5!*x^n. Cf. A001725.

%F A(x)/(1 - 6*x*A(x)) = Sum_{n >= 0} (n+6)!/6!*x^n. Cf. A001730.

%F A(x) satisfies the Riccati equation x^2*A'(x) + 6*x*A^2(x) - (1 + 5*x)*A(x) + 1 = 0.

%F G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 7*x/(1 - 2*x/(1 - 8*x/(1 - 3*x/(1 - 9*x/(1 - ... - n*x/(1 - (n+6)*x/(1 - ... ))))))))), by Stokes 1982.

%F A(x) = 1/(1 + 6*x - 7*x/(1 - x/(1 - 8*x/(1 - 2*x/(1 - 9*x/(1 - 3*x/(1 - ... - (n + 6)*x/(1 - n*x/(1 - ... ))))))))). (End)

%e (1/6)*(log(1 + 6*x + 42*x^2 + 336*x^3 + ... + (n+5)!/5!)*x^n + ...)

%e = x + 8/2*x^2 + 78/3*x^3 + 876/4*x^4 + 10956/5*x^5 + ...

%t m = 18; (-1/(6x)) ContinuedFractionK[-i x, 1 + i x, {i, 6, m+5}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 02 2019 *)

%o (PARI) {a(n)=if(n<0,0,if(n==0,1, (n/6)*polcoeff(log(sum(m=0,n,(m+5)!/5!*x^m) + x*O(x^n)),n)))} \\ fixed by _Vaclav Kotesovec_, Jul 27 2015

%Y Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111534 (diagonal).

%Y Cf. A001725, A001730.

%K nonn,easy

%O 0,3

%A _Paul D. Hanna_, Aug 06 2005