A045894 4-fold convolution of A001700(n), n >= 0.

1, 12, 94, 608, 3525, 19044, 97954, 486000, 2345930, 11081880, 51447036, 235454848, 1064832173, 4767347796, 21160397050, 93223960784, 408037319262, 1775744775592, 7688699122724, 33140226601920, 142262721338146

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1500
• José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
• Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Crossrefs

Cf. A001700, A000108, A045720.

Programs

Mathematica

Table[(n + 11)*4^(n + 2) - (n + 5) Binomial[2 (n + 4), n + 4]/2, {n, 0, 20}] (* Michael De Vlieger, Feb 18 2017 *)

Python

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A045894(n):
    return (n+11)*4**(n+2)-(n+5)*C(2*(n+4),(n+4))/2 # Indranil Ghosh, Feb 18 2017

Formula

a(n) = (n+11)*4^(n+2) - (n+5)*binomial(2*(n+4), n+4)/2;

G.f.: c(x)^4/(1-4*x)^2, where c(x) = g.f. for Catalan numbers A000108;

recursion: a(n)= (2*(2*n+10)/(n+4))*a(n-1) + (4/(n+4))*A045720(n), a(0)=1.