A176604 Sequence defined by the recurrence formula a(n+1) = Sum_{n>=1}(a(p)*a(n-p) + k, p=0..n) + j, with a(0) = 1, a(1) = 0, k = 0 and j = 1.

1, 0, 1, 3, 7, 16, 39, 102, 279, 782, 2227, 6427, 18769, 55376, 164801, 494071, 1490663, 4522690, 13790171, 42234621, 129866725, 400765128, 1240796725, 3853055776, 11997619209, 37451945874, 117181432493, 367428949069

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

Sequence in context: A152090 A190528 A203611 * A014140 A271788 A103439

Adjacent sequences: A176601 A176602 A176603 * A176605 A176606 A176607

Keyword

easy,nonn

Author

Richard Choulet, Apr 21 2010

Status

approved