A112700 Partial sum of Catalan numbers A000108 multiplied by powers of 6.

1, 7, 79, 1159, 19303, 345895, 6504487, 126597031, 2528447911, 51526205863, 1067116097959, 22394503831975, 475191351108007, 10177980935594407, 219758235960500647, 4778128782752211367, 104526001924311998887

Offset: 0

Author

Wolfdieter Lang, Oct 31 2005

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 0..727

Crossrefs

Seventh column (m=6) of triangle A112705.

Programs

Maple

ListTools:-PartialSums([seq(binomial(2*n,n)/(n+1)*6^n,n=0..50)]); # Robert Israel, Jun 28 2018

Mathematica

Accumulate[Table[ (CatalanNumber@ n)*6^n, {n, 0, 16}]] (* James C. McMahon, Jun 11 2024 *)

Formula

a(n) = Sum_{k=0..n} C(k)*6^k, with C(n):=A000108(n).

G.f.: c(6*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.

Conjecture: (n+1)*a(n) +(-25*n+11)*a(n-1) +12*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 08 2016, verified by Robert Israel, Jun 28 2018

0 = a(n)*(+576*a(n+1) -636*a(n+2) +60*a(n+3)) +a(n+1)*(-564*a(n+1) +613*a(n+2) -61*a(n+3)) +a(n+2)*(+11*a(n+2) +a(n+3)) for all n>=0. - Michael Somos, Jun 28 2018