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A093388 (n+1)^2*a(n+1) = (17n^2+17n+6)*a(n) - 72*n^2*a(n-1). 2
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

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

LINKS

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

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982.

Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.

Matthijs Coster, Sequences

Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

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

H. Verrill, Some congruences related to modular forms, Section 2.2.

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

0 = x*(x+8)*(x+9)*y'' + (3*x^2 + 34*x + 72)*y' + (x+6)*y, where y(x) = A(x/-72). - Gheorghe Coserea, Aug 26 2016

EXAMPLE

A(x) = 1 + 6*x + 42*x^2 + 312*x^3 + 2394*x^4 + 18756*x^5 + ... is the g.f.

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));

(PARI)

seq(N) = {

  my(a = vector(N)); a[1] = 6; a[2] = 42;

  for (n=3, N, a[n] = ((17*n^2 - 17*n + 6)*a[n-1] - 72*(n-1)^2*a[n-2])/n^2);

  concat(1, a);

};

seq(20)  \\ Gheorghe Coserea, Aug 26 2016

CROSSREFS

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

Cf. A091401.

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 December 9 14:26 EST 2016. Contains 278971 sequences.