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 A111531 Row 4 of table A111528. 14
 1, 1, 6, 46, 416, 4256, 48096, 591536, 7840576, 111226816, 1680157056, 26918720896, 455971214336, 8143926373376, 153013563734016, 3017996904928256, 62369444355076096, 1348096649995841536, 30426167700424728576, 715935203128235401216 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..440 A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214. FORMULA G.f.: (1/4)*Log(Sum_{n>=0} (n+3)!/3!*x^n) = Sum_{n>=1} a(n)*x^n/n. G.f.: A(x) = 1/(1+4*x - 5*x/(1+5*x - 6*x/(1+6*x -... (continued fraction). a(n)=Sum_{k, 0<=k<=n}4^(n-k)*A089949(n,k) . - Philippe Deléham, Oct 16 2006 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 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 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 From Peter Bala, May 25 2017: (Start) O.g.f. A(x) = ( Sum_{n >= 0} (n+4)!/4!*x^n ) / ( Sum_{n >= 0} (n+3)!/3!*x^n ). 1/(1 - 4*x*A(x)) = Sum_{n >= 0} (n+3)!/3!*x^n. Cf. A001715. A(x)/(1 - 4*x*A(x)) = Sum_{n >= 0} (n+4)!/4!*x^n. Cf. A001720. A(x) satisfies the Riccati equation x^2*A'(x) + 4*x*A^2(x) - (1 + 3*x)*A(x) + 1 = 0 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. 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) EXAMPLE (1/4)*Log(1 + 4*x + 20*x^2 + 120*x^3 +... + (n+3)!/3!*x^n +...) = x + 6/2*x^2 + 46/3*x^3 + 416/4*x^4 + 4256/5*x^5 +... PROG (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)))} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111532 (row 5), A111533 (row 6), A111534 (diagonal). Cf. A001715, A001720. Sequence in context: A271933 A084772 A199563 * A052781 A284109 A049378 Adjacent sequences:  A111528 A111529 A111530 * A111532 A111533 A111534 KEYWORD nonn,easy,changed AUTHOR Paul D. Hanna, Aug 06 2005 STATUS approved

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