OFFSET
0,4
COMMENTS
The link contains a list of all 85 related sequences and their parameters a(1)=m, k and j, together with a proof of the recurrence given by Richard J. Mathar. - Georg Fischer, Jan 26 2020
LINKS
Georg Fischer, Table of n, a(n) for n = 0..200
Georg Fischer, Derivation of the D-finite recurrence equation for A176604 and related sequences.
FORMULA
G.f.: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+j)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, j=1).
(n+1)*a(n) +2*(-3*n+1)*a(n-1) +(13*n-21)*a(n-2) +16*(-n+3)*a(n-3) +8*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 19 2016
EXAMPLE
a(2) = (a(0)*a(1)+0)+(a(1)*a(0)+0)+1 = 1.
a(3) = (a(0)*a(2)+0)+(a(1)^2+0)+(a(2)*a(0)+0)+1 = 3.
a(4) = 2*a(0)*a(3)+2*a(1)*a(2)+1 = 7.
MATHEMATICA
(* Applicable for all 85 related sequences *)
m:=0; k:=0; j:=1; CoefficientList[Series[(1-Sqrt[1-4*z*(1-z+z*m+(k+j)*z^2/(
1-z)+k*z^2/(1-z)^2)])/(2*z), {z, 0, 20}], z]
(* or *)
m:=0; k:=0; j:=1;
RecurrenceTable[{a[0]==1, a[1]==m, a[2]==1, a[3]==3, a[4]==7,
+ (+20+20*k+20*j-20*m+(- 4- 4*k-4*j+ 4*m)*n)*a[n-5]
- (+62+48*k+34*j-48*m+(-16-12*k-8*j+12*m)*n)*a[n-4]
+ (+68+28*k+14*j-36*m+(-25- 8*k-4*j+12*m)*n)*a[n-3]
- (+29 - 8*m+(-19 + 4*m)*n)*a[n-2]
- ( -2 +( 7 )*n)*a[n-1]
+ ( +1 +( 1 )*n)*a[n ]
== 0}, a, {n, 0, 20}]== 0}, a, {n, 0, 20}] (* Georg Fischer, Jan 26 2020 - cf. link *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Richard Choulet, Apr 21 2010
STATUS
approved