A112701 Partial sum of Catalan numbers A000108 multiplied by powers of 7.

1, 8, 106, 1821, 35435, 741329, 16270997, 369570944, 8613236374, 204812473608, 4949266755812, 121188396669810, 3000342229924222, 74979188061284522, 1888846103011564082, 47915719069874907917, 1222954711282739097587

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..693

Formula

a(n) = Sum_{k=0..n} C(k)*7^k, n>=0, with C(n):=A000108(n).

G.f.: c(7*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) +(-29*n+13)*a(n-1) +14*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 08 2016

Conjecture verified using the d.e. (28*x^3-29*x^2+x)*y' + (42*x^2-16*x+1)*y=1 satisfied by the g.f. - Robert Israel, Aug 04 2020

Maple

f:= gfun:-rectoproc({(n+1)*a(n) +(-29*n+13)*a(n-1) +14*(2*n-1)*a(n-2)=0, a(0)=1, a(1)=8}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Aug 04 2020

Mathematica

CatalanNumber[#]*7^#& /@ Range[0, 20] // Accumulate (* Jean-François Alcover, Aug 29 2022 *)

Crossrefs

Eighth column (m=7) of triangle A112705.

Sequence in context: A155632 A129278 A369512 * A380258 A099695 A236953

Adjacent sequences: A112698 A112699 A112700 * A112702 A112703 A112704

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 31 2005

Status

approved