Derivation of the D-finite recurrence equation 
for A176604 and related sequences
==============================================
Georg Fischer, Jan 26 2020

There are 85 sequences in the range A176604-A177203
whose names contain essentially 3 parameters
j, k and a(1)=m (a(0) is always 1):

  Sequence defined by the recurrence formula 
  a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+j for n>=1, 
  with here a(0)=1, a(1)=0, k=0 and j=1.
  
For the general case with arbitrary j, k and m we show 
the derivation of the D-finite recurrence by means of 
a Mathematica session:

We fetch the package of Wolfram Koepf from
<http://www.mathematik.uni-kassel.de/~koepf/Publikationen/SpecialFunctions.m>,
store it in the local Mathematica directory structure,
and start a fresh MMA session. 

  <<SpecialFunctions`

We rewrite the formula for the g.f. by expanding the fraction 
with (1-x), since the SpecialFunctions package crashes with the 
original formula.

  gf=(-Sqrt[1-6*x+(13-4*m)*x^2+(-12-8*k-4*j+8*m)*x^3+(4+4*k+4*j-4*m)*x^4])/(2*x*(1-x));

For this g.f. compute the differential equation:
  de=HolonomicDE[gf,f[x]]

which is:
  (-1+5*x-9*x^2+(7-4*k-2*j)*x^3+(-2+2*j)*x^4) * f[x] - 
  (-1+x)*x*(-1+6*x+(-13+4*m)*x^2+(12+8*k+4*j-8*m)*x^3+(-4-4*k-4*j+4*m)*x^4) * f'[x] == 0

With
  re=DEtoRE[de,f[x],a[n]]

we get:
 4 (1 + k + j - m) n a[n] 
 + 2 (-9 - 6 k - 3 j + 6 m - 8 n - 6 k n - 4 j n + 6 m n) a[1 + n] 
 - (-57 - 12 k - 6 j + 24 m - 25 n - 8 k n - 4 j n + 12 m n) a[2 + n] 
 + (-66 + 12 m - 19 n + 4 m n) a[3 + n] 
 + (33 + 7 n) a[4 + n] 
 - (6 + n) a[5 + n] == 0

Here we shift n to n-5, initialize a[2..4]
and finally get the following program:

(* A176604: a(1)=0, k=0 and j=1. 
   1, 0, 1, 3, 7, 16, 39, 102, 279, 782, 2227, 6427, 18769, 55376 *)
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}]

-----------------------

In the special case of A176604, we see that the differential 
equation has a common factor (x-1) which can be cancelled out:

  m=0; j=1; k=0;
  Factor[  (-1+5*x-9*x^2+(7-4*k-2*j)*x^3+(-2+2*j)*x^4) * f[x] -
    (-1+x)*x*(-1+6*x+(-13+4*m)*x^2+(12+8*k+4*j-8*m)*x^3+(-4-4*k-4*j+4*m)*x^4) * f'[x]
    == 0]
    
  (-1 + x)*((1-- 4*x + 5*x^2 - 2*x^3 + 2*j*x^3)*f[x] +
   (x - 6*x^2 + 13*x^3 - 12*x^4 - 4*j*x^4 + 4*x^5 + 4*j*x^5)*f'[x]) == 0
  
  de=((1-4*x...)*f[x] + (x-6*x^2 ... )*f'[x]) == 0
  re=DEtoRE[de,f[x],a[n]]
  
  4*(1+j)*n*a[n]
  -2*(7+j+6*n+2*j*n)*a[1+n]
  +(31+13*n)*a[2+n]
  -2*(11+3*n)*a[3+n]
  +(5+n)*a[4+n] == 0

By shifting n -> n-4 we get:

  8*(n-4)*a[n-4]-16(n-3)a[n-3]+(13n-21))*a[n-2]-2(3n-1))*a[n-1]+(n+1)*a[n]==0

which is exactly the recurrence given by Richard J. Mathar:

  (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 

-----------------------

By setting k, j and m and a[2..4], all remaining 84
sequences can be computed.

The following list shows their parameters:

A-number  a(0)  m   k   j
------------------------- 
A176604     1   0   0   1
A176605     1   1   0   1
A176606     1   3   0   1
A176607     1   4   0   1
A176609     1   5   0   1
A176610     1   0   1   1
A176611     1   1   1   1
A176612     1   2   1   1
A176645     1   4   1   1
A176648     1   5   1   1
A176675     1   0   0   -1
A176677     1   1   0   -1
A176678     1   2   0   -1
A176749     1   3   0   -1
A176750     1   4   0   -1
A176751     1   5   0   -1
A176752     1   0   0   -2
A176753     1   1   0   -2
A176754     1   2   0   -2
A176755     1   3   0   -2
A176756     1   4   0   -2
A176757     1   5   0   -2
A176759     1   0   1   -1
A176828     1   2   1   -1
A176829     1   3   1   -1
A176830     1   4   1   -1
A176832     1   5   1   -1
A176854     1   0   -1  0
A176855     1   1   -1  0
A176856     1   2   -1  0
A176857     1   3   -1  0
A176858     1   4   -1  0
A176859     1   5   -1  0
A176952     1   0   -1  -1
A176953     1   1   -1  -1
A176954     1   2   -1  -1
A176956     1   3   -1  -1
A176957     1   4   -1  -1
A176958     1   5   -1  -1
A176959     1   0   -1  1
A176962     1   2   -1  1
A176964     1   3   -1  1
A176966     1   4   -1  1
A176967     1   5   -1  1
A177110     1   0   -2  0
A177111     1   1   -2  0
A177113     1   2   -2  0
A177115     1   3   -2  0
A177117     1   4   -2  0
A177118     1   5   -2  0
A177122     1   6   1   1
A177123     1   7   1   1
A177124     1   8   1   1
A177125     1   9   1   1
A177126     1   10  1   1
A177127     1   6   0   1
A177128     1   7   0   1
A177129     1   8   0   1
A177130     1   9   0   1
A177131     1   10  0   1
A177162     1   6   0   -1
A177163     1   7   0   -1
A177165     1   8   0   -1
A177166     1   9   0   -1
A177167     1   10  0   -1
A177168     1   6   0   -2
A177169     1   7   0   -2
A177170     1   8   0   -2
A177171     1   9   0   -2
A177172     1   10  0   -2
A177175     1   6   1   -1
A177177     1   7   1   -1
A177178     1   8   1   -1
A177179     1   9   1   -1
A177180     1   10  1   -1
A177181     1   6   -1  -1
A177182     1   7   -1  -1
A177183     1   8   -1  -1
A177184     1   9   -1  -1
A177185     1   10  -1  -1
A177197     1   6   -1  1
A177198     1   7   -1  1
A177199     1   8   -1  1
A177200     1   9   -1  1
A177203     1   10  -1  1