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
Links
- Robert Israel, Table of n, a(n) for n = 0..693
Programs
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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 *)
Formula
a(n) = Sum_{k=0..n} A000108(k)*7^k.
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
a(n) = 7^n*binomial(2*n, n)*(1 - hypergeom([1, n+1/2], [n+2], 28))/(n+1) + (1 - 3*sqrt(3)*i)/14, where i denotes the imaginary units. - Stefano Spezia, Mar 31 2025