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

0,2

Links

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

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

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 *)

Crossrefs

Seventh column (m=6) of triangle A112705.

Sequence in context: A014293 A176792 A186377 * A365039 A375173 A365782

Adjacent sequences: A112697 A112698 A112699 * A112701 A112702 A112703

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 31 2005

Status

approved