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A093388 (n+1)^2 a_{n+1} = (17n^2+17n+6) a_n - 72n^2 a_{n-1}. 1
1, 6, 42, 312, 2394, 18756, 149136, 1199232, 9729882, 79527084, 654089292, 5408896752, 44941609584, 375002110944, 3141107339328, 26402533581312, 222635989516122, 1882882811380284, 15967419789558804, 135752058036988848, 1156869080242393644 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the Taylor expansion of a special point on a curve described by Beauville.

REFERENCES

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.

Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, http://www.carma.newcastle.edu.au/~jb616/walks.pdf, Random Walk Integrals, 2010. - From N. J. A. Sloane, Feb 22 2013

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From N. J. A. Sloane, Dec 16 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Matthijs Coster, Sequences

H. Verrill, Some congruences related to modular forms, Section 2.2. [broken link]

FORMULA

(-1)^n * sum_{k=0}^n binomial(n, k) * (-8)^k * sum_{j=0}^{n-k} binomial(n-k, j)^3 - Helena Verrill (verrill(AT)math.lsu.edu), Aug 09 2004

G.f.: hypergeom([1/3, 2/3],[1],x^2*(8*x-1)/(2*x-1/3)^3)/(1-6*x)  - Mark van Hoeij, Oct 25 2011.

a(n) ~ 3^(2*n+3/2)/(Pi*n). - Vaclav Kotesovec, Oct 14 2012

MAPLE

f:=proc(n) option remember; local m; if n=0 then RETURN(1); fi; if n=1 then RETURN(6); fi; m:=n-1; ((17*m^2+17*m+6)*f(n-1)-72*m^2*f(n-2))/n^2; end;

MATHEMATICA

Table[(-1)^n*Sum[Binomial[n, k]*(-8)^k*Sum[Binomial[n-k, j]^3, {j, 0, n-k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PROG

(PARI) a(n)=(-1)^n*sum(k=0, n, binomial(n, k)*(-8)^k*sum(j=0, n-k, binomial(n-k, j)^3));

CROSSREFS

This is the seventh sequence in the family beginning A002894, A006077, A081085, A005258, A000172, A002893.

Sequence in context: A111602 A091164 A004982 * A162968 A247638 A034171

Adjacent sequences:  A093385 A093386 A093387 * A093389 A093390 A093391

KEYWORD

nonn

AUTHOR

Matthijs Coster, Apr 29 2004

STATUS

approved

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Last modified July 28 22:54 EDT 2015. Contains 260099 sequences.