A111533 Row 6 of table A111528.

1, 1, 8, 78, 876, 10956, 149472, 2195208, 34398288, 571525200, 10022997888, 184897670112, 3578224662720, 72486450479808, 1534267158087168, 33877135427154048, 779208751651730688, 18645519786163266816

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

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

a(n) = Sum_{k=0..n} 6^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006

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

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

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

From Peter Bala, May 25 2017: (Start)

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

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

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

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

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.

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)

Example

(1/6)*(log(1 + 6*x + 42*x^2 + 336*x^3 + ... + (n+5)!/5!)*x^n + ...)
= x + 8/2*x^2 + 78/3*x^3 + 876/4*x^4 + 10956/5*x^5 + ...

Mathematica

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 *)

Prog

(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

Crossrefs

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

Cf. A001725, A001730.

Sequence in context: A155600 A091106 A199758 * A218499 A132164 A058480

Adjacent sequences: A111530 A111531 A111532 * A111534 A111535 A111536

Keyword

nonn,easy

Author

Paul D. Hanna, Aug 06 2005

Status

approved