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A111532 Row 5 of table A111528.

%I A111532 #35 Jan 21 2020 00:04:49
%S A111532 1,1,7,61,619,7045,87955,1187845,17192275,264940405,4326439075,
%T A111532 74593075525,1353928981075,25809901069525,515683999204675,
%U A111532 10779677853137125,235366439343773875,5359766538695291125
%N A111532 Row 5 of table A111528.
%H A111532 Vaclav Kotesovec, <a href="/A111532/b111532.txt">Table of n, a(n) for n = 0..440</a>
%H A111532 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 A111532 G.f.: (1/5)*log(Sum_{n>=0} (n+4)!/4!*x^n) = Sum_{n>=1} a(n)*x^n/n.
%F A111532 G.f.: 1/(1 + 5*x - 6*x/(1 + 6*x - 7*x/(1 + 7*x - ... (continued fraction).
%F A111532 a(n) = Sum_{k=0..n} 5^(n-k)*A089949(n,k). - _Philippe Deléham_, Oct 16 2006
%F A111532 G.f.: (4 + 1/Q(0))/5, where Q(k) = 1 - 3*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 04 2013
%F A111532 a(n) ~ n! * n^5/5! * (1 + 5/n - 55/n^3 - 356/n^4 - 3095/n^5 - 35225/n^6 - 475000/n^7 - 7293775/n^8 - 124710375/n^9 - 2339428250/n^10). - _Vaclav Kotesovec_, Jul 27 2015
%F A111532 From _Peter Bala_, May 25 2017: (Start)
%F A111532 O.g.f.: A(x) = ( Sum_{n >= 0} (n+5)!/5!*x^n ) / ( Sum_{n >= 0} (n+4)!/4!*x^n ).
%F A111532 1/(1 - 5*x*A(x)) = Sum_{n >= 0} (n+4)!/4!*x^n. Cf. A001720.
%F A111532 A(x)/(1 - 5*x*A(x)) = Sum_{n >= 0} (n+5)!/5!*x^n. Cf. A001725.
%F A111532 A(x) satisfies the Riccati equation x^2*A'(x) + 5*x*A^2(x) - (1 + 4*x)*A(x) + 1 = 0.
%F A111532 G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 3*x/(1 - 8*x/(1 - ... - n*x/(1 - (n+5)*x/(1 - ... ))))))))), by Stokes 1982.
%F A111532 A(x) = 1/(1 + 5*x - 6*x/(1 - x/(1 - 7*x/(1 - 2*x/(1 - 8*x/(1 - 3*x/(1 - ... - (n + 5)*x/(1 - n*x/(1 - ... ))))))))). (End)
%e A111532 (1/5)*(log(1 + 5*x + 30*x^2 + 210*x^3 + ... + (n+4)!/4!)*x^n + ...)
%e A111532 = x + 7/2*x^2 + 61/3*x^3 + 619/4*x^4 + 7045/5*x^5 + ...
%t A111532 m = 18; (-1/(5x)) ContinuedFractionK[-i x, 1 + i x, {i, 5, m+4}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 02 2019 *)
%o A111532 (PARI) {a(n)=if(n<0,0,if(n==0,1, (n/5)*polcoeff(log(sum(m=0,n,(m+4)!/4!*x^m) + x*O(x^n)),n)))} \\ fixed by _Vaclav Kotesovec_, Jul 27 2015
%Y A111532 Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111531 (row 4), A111533 (row 6), A111534 (diagonal).
%Y A111532 Cf. A001720, A001725.
%K A111532 nonn,easy
%O A111532 0,3
%A A111532 _Paul D. Hanna_, Aug 06 2005