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A111531 Row 4 of table A111528.

%I A111531 #37 Jan 21 2020 00:04:45
%S A111531 1,1,6,46,416,4256,48096,591536,7840576,111226816,1680157056,
%T A111531 26918720896,455971214336,8143926373376,153013563734016,
%U A111531 3017996904928256,62369444355076096,1348096649995841536,30426167700424728576,715935203128235401216
%N A111531 Row 4 of table A111528.
%H A111531 Vaclav Kotesovec, <a href="/A111531/b111531.txt">Table of n, a(n) for n = 0..440</a>
%H A111531 Paul Barry, <a href="https://arxiv.org/abs/1804.06801">A note on number triangles that are almost their own production matrix</a>, arXiv:1804.06801 [math.CO], 2018.
%H A111531 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 A111531 G.f.: (1/4)*log(Sum_{n>=0} (n+3)!/3!*x^n) = Sum_{n>=1} a(n)*x^n/n.
%F A111531 G.f.: A(x) = 1/(1 + 4*x - 5*x/(1 + 5*x - 6*x/(1 + 6*x - ... (continued fraction).
%F A111531 a(n) = Sum_{k=0..n} 4^(n-k)*A089949(n,k). - _Philippe Deléham_, Oct 16 2006
%F A111531 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 05 2013
%F A111531 G.f.: W(0)/4 + 3/4, where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+2)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Aug 26 2013
%F A111531 a(n) ~ n! * n^4/24 * (1 + 2/n - 5/n^2 - 30/n^3 - 184/n^4 - 1664/n^5 - 18688/n^6 - 245120/n^7 - 3641280/n^8 - 60090368/n^9 - 1086985152/n^10). - _Vaclav Kotesovec_, Jul 27 2015
%F A111531 From _Peter Bala_, May 25 2017: (Start)
%F A111531 O.g.f. A(x) = ( Sum_{n >= 0} (n+4)!/4!*x^n ) / ( Sum_{n >= 0} (n+3)!/3!*x^n ).
%F A111531 1/(1 - 4*x*A(x)) = Sum_{n >= 0} (n+3)!/3!*x^n. Cf. A001715.
%F A111531 A(x)/(1 - 4*x*A(x)) = Sum_{n >= 0} (n+4)!/4!*x^n. Cf. A001720.
%F A111531 A(x) satisfies the Riccati equation x^2*A'(x) + 4*x*A^2(x) - (1 + 3*x)*A(x) + 1 = 0.
%F A111531 G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 3*x/(1 - 7*x/(1 - ... - n*x/(1 - (n+4)*x/(1 - ... ))))))))), by Stokes 1982.
%F A111531 A(x) = 1/(1 + 4*x - 5*x/(1 - x/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 3*x/(1 - ... - (n + 4)*x/(1 - n*x/(1 - ... ))))))))). (End)
%e A111531 (1/4)*(log(1 + 4*x + 20*x^2 + 120*x^3 + ... + (n+3)!/3!)*x^n + ...)
%e A111531 = x + 6/2*x^2 + 46/3*x^3 + 416/4*x^4 + 4256/5*x^5 + ...
%t A111531 T[n_, k_] := T[n, k] = Which[n<0 || k<0, 0, k==0 || k==1, 1, n==0, k!, True, (T[n-1, k+1]-T[n-1, k])/n-Sum[T[n, j]*T[n-1, k-j], {j, 1, k-1}]];
%t A111531 a[n_] := T[4, n];
%t A111531 a /@ Range[0, 19] (* _Jean-François Alcover_, Oct 01 2019 *)
%o A111531 (PARI) {a(n)=if(n<0,0,if(n==0,1, (n/4)*polcoeff(log(sum(m=0,n,(m+3)!/3!*x^m) +x*O(x^n)),n)))}
%o A111531 for(n=0,20,print1(a(n),", "))
%Y A111531 Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111532 (row 5), A111533 (row 6), A111534 (diagonal).
%Y A111531 Cf. A001715, A001720.
%K A111531 nonn,easy
%O A111531 0,3
%A A111531 _Paul D. Hanna_, Aug 06 2005