|
| |
|
|
A181167
|
|
G.f.: 1 = Sum_{n>=0} a(n)*x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.
|
|
4
|
|
|
|
1, 1, 8, 165, 6384, 397320, 36273600, 4566166605, 757975618400, 160424015864112, 42164387189608320, 13473505313334666600, 5144136790654611953280, 2312696796696904699224000, 1209297981696245764641077760, 727688337054213932985609546525
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Compare g.f. to a g.f of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n * Sum_{k>=0} C(2n+k,k)*(-x)^k.
|
|
|
LINKS
|
|
|
|
FORMULA
|
Differentiating -log(BesselJ(0,2*sqrt(x))) and substituting z=z^2 gives the e.g.f. Sum_{n>=0} a(n) * z^(2*n)/(2n)! = BesselJ(1,2*z)/z/BesselJ(0,2*z). Consequently, using Gauss' continued fraction, this e.g.f. is also: 1/(1-z^2/(2-z^2/(3-z^2/(4-z^2/(5-z^2/...))))). - Matthieu Josuat-Vergès, Apr 17 2011
E.g.f.: U(0) where U(k) = 1 - x^2/(x^2 - (k+1)*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 15 2012
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 16/BesselJZero[0,1]^2 = 2.76664110449031883070186935..., c = 4/(sqrt(Pi)*BesselJZero[0,1]^2) = 0.390227523142124366836071453... . - Vaclav Kotesovec, Jul 31 2014
|
|
|
EXAMPLE
|
E.g.f.: E(x) = 1 + x^2/2! + 8*x^4/4! + 165*x^6/6! + 6384*x^8/8! +...
where the e.g.f. equals the continued fraction:
E(x) = 1/(1 - x^2/(2 - x^2/(3 - x^2/(4 - x^2/(5 - x^2/(6 - x^2/(7 - x^2/(8 - x^2/(9 - x^2/(10 - ...)))))))))). [Due to Matthieu Josuat-Vergès]
Illustrate the g.f. by the series:
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 - 21^2*x^5 +...)
+ 8*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 - 126^2*x^5 +...)
+ 165*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 - 462^2*x^5+...)
+ 6384*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 397320*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 36273600*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 4566166605*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4 +...)
+...
Compare to a g.f. of the Catalan numbers (A000108):
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3*x + 6*x^2 - 10*x^3 + 15*x^4 - 21*x^5 +...)
+ 2*x^2*(1 - 5*x + 15*x^2 - 35*x^3 + 70*x^4 - 126*x^5 +...)
+ 5*x^3*(1 - 7*x + 28*x^2 - 84*x^3 + 210*x^4 - 462*x^5 +...)
+ 14*x^4*(1 - 9*x + 45*x^2 - 165*x^3 + 495*x^4 - 1287*x^5 +...)
+ 42*x^5*(1 - 11*x + 66*x^2 - 286*x^3 + 1001*x^4 - 3003*x^5 +...)
+ 132*x^6*(1 - 13*x + 91*x^2 - 455*x^3 + 1820*x^4 - 6188*x^5 +...)
+...
Surprisingly, terms a(n) are divisible by n*A000108(n) for n>0:
a(2)=2*2*2, a(3)=3*5*11, a(4)=4*14*114, a(5)=5**42*1892, a(6)=6*132*45800, a(7)=7*429*1520535, ..., a(n)=n*A000108(n)*A181168(n).
|
|
|
MAPLE
|
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, x/(y+1)*(b(x-1, y-1)+b(x-1, y+1))))
end:
a:= n-> b(2*n, 0):
|
|
|
MATHEMATICA
|
nmax=20; Table[(CoefficientList[Series[BesselJ[1, 2*x]/x/BesselJ[0, 2*x], {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n-1]], {n, 1, nmax}] (* Vaclav Kotesovec, Jul 31 2014 *)
|
|
|
PROG
|
(PARI) {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, n-m, binomial(2*m+k, k)^2*(-x)^k)+x*O(x^n)), n))}
{a(n)=binomial(2*n, n)/(n+1)*(n+1)!^2*polcoeff(-log(sum(m=0, n+1, (-x)^m/m!^2)+O(x^(n+2))), n+1)} (End)
(PARI) /* Continued Fraction expansion of the E.G.F.: */
{a(n)=local(CF=1+O(x)); for(i=0, n, CF=1/((n-i+1)-x^2*CF)); (2*n)!*polcoeff(CF, 2*n)}
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
