A018218 Sum(C(j)*(n-j)*4^(n-j-1),j=0..n-1), C = Catalan numbers.

0, 1, 9, 58, 325, 1686, 8330, 39796, 185517, 848830, 3827230, 17053356, 75249954, 329353948, 1431575220, 6185613032, 26589395581, 113780713806, 484945025942, 2059546425340, 8719018250838, 36805967321684

Offset: 0

Author

N. J. A. Sloane, Peter Winkler (pw(AT)bell-labs.com)

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..173

Programs

Magma

[(n+1)*(4^n-Binomial(2*n+1, n))/2: n in [0..25]]; // Vincenzo Librandi, Jun 09 2011

Mathematica

Table[Sum[CatalanNumber[j](n-j)4^(n-j-1),{j,-0,n-1}],{n,0,30}] (* Harvey P. Dale, Nov 15 2020 *)

Formula

a(n)=(n+1)*(4^n-binomial(2*n+1, n))/2; G.f.: x*c(x)/(1-4*x)^2, where c(x) = g.f. for Catalan numbers A000108; also convolution of A000346(n-1), n >= 0, where A000346(-1)=0, with A000302 (powers of 4). - Wolfdieter Lang

Asymptotics: a(n) ~ 2^(2*n-1)*(n+1-sqrt(4*n/Pi)). - Fung Lam, Mar 28 2014

Recurrence: (n-1)*n*a(n) = 2*(n-1)*(4*n+1)*a(n-1) - 8*n*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 28 2014