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

0,3

Links

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

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

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A027174 A304370 A099624 * A026750 A009034 A026377

Adjacent sequences: A018215 A018216 A018217 * A018219 A018220 A018221

Keyword

nonn

Author

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

Status

approved