A050253 G.f.: ( 1 - x^2 - sqrt( 1 - 2*x^2 - 4*x^3 - 3*x^4 ) ) / ( 2*x^3 ).
1, 1, 1, 2, 3, 5, 9, 16, 29, 54, 101, 191, 365, 702, 1359, 2647, 5181, 10187, 20113, 39856, 79243, 158036, 316053, 633689, 1273559, 2565136, 5177043, 10468199, 21204379, 43022215, 87423573, 177906552, 362531425, 739700055, 1511091377
Offset: 0
Links
- Robert Israel, Table of n, a(n) for n = 0..3056
Programs
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Maple
f:= gfun:-rectoproc({(3+3*n)*a(n)+(10+4*n)*a(1+n)+(2*n+8)*a(n+2)+(-7-n)*a(n+4), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2}, a(n), remember): map(f, [$0..50]); # Robert Israel, Jan 15 2018 -
Mathematica
CoefficientList[Series[(1-x^2-Sqrt[1-2x^2-4x^3-3x^4])/(2x^3),{x,0,40}],x] (* Harvey P. Dale, Jul 17 2015 *) -
Maxima
a(n):=if n=0 then 1 else sum(((sum(binomial(j,n-k-j)*binomial(k,j),j,0,k))*binomial(n-k,k-1))/k,k,1,n); /* Vladimir Kruchinin, Nov 21 2014 */
Formula
G.f.: 1/(1-x-x^3/(1-x^2-x^3/(1-x-x^3/(1-x^2-x^3/(1-x-x^3/(1-... (continued fraction). - Paul Barry, May 25 2009
G.f.: 1/(1-x/(1-x^2/(1-x^3/(1-x/(1-x^2/(1-x^3/(1-x/(1-x^2/(1-x^3/(1-... (continued fraction). - Paul Barry, Jul 29 2010
D-finite with recurrence: (n+3)*a(n) + (n+2)*a(n-1) - 2n*a(n-2) + 2*(4-3n)*a(n-3) + (19-7n)*a(n-4) + 3*(4-n)*a(n-5) = 0. - R. J. Mathar, Nov 15 2011
From Robert Israel, Jan 15 2018: (Start)
Recurrence verified using the differential equation (3*x^5+4*x^4+2*x^3-x)*y' + (3*x^4+6*x^3+4*x^2-3)*y + x^2+4*x+3 = 0 satisfied by the g.f.
(3+3*n)*a(n) + (10+4*n)*a(1+n) + (2*n+8)*a(n+2) + (-7-n)*a(n+4) = 0. (End)
a(n) = Sum_{k=1..n} (Sum_{j=0..k} C(j,n-k-j)*C(k,j))*C(n-k,k-1)/k for n > 0. - Vladimir Kruchinin, Nov 21 2014
G.f. A(x) satisfies A(x) = x*(1+sqrt(1+4*(A(x)+A(x)^2+A(x)^3)))/2. - Vladimir Kruchinin, Nov 21 2014
a(0) = a(1) = 1; a(n) = a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Apr 11 2021
a(n) ~ sqrt(6 + 3*r - 6*r^2) * (6 + 5*r + 6*r^2) * (1 + r + 3*r^2)^n / (6*sqrt(Pi)*n^(3/2)), where r = 0.4693964245699946792019209673920017843813793... is the root of the equation 3*r^3 + r^2 + r - 1 = 0. - Vaclav Kotesovec, Jul 03 2021
Comments