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%I #14 Aug 29 2022 10:22:25
%S 1,8,106,1821,35435,741329,16270997,369570944,8613236374,204812473608,
%T 4949266755812,121188396669810,3000342229924222,74979188061284522,
%U 1888846103011564082,47915719069874907917,1222954711282739097587
%N Partial sum of Catalan numbers A000108 multiplied by powers of 7.
%H Robert Israel, <a href="/A112701/b112701.txt">Table of n, a(n) for n = 0..693</a>
%F a(n) = Sum_{k=0..n} C(k)*7^k, n>=0, with C(n):=A000108(n).
%F 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.
%F 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
%F 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
%p 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):
%p map(f, [$0..50]); # _Robert Israel_, Aug 04 2020
%t CatalanNumber[#]*7^#& /@ Range[0, 20] // Accumulate (* _Jean-François Alcover_, Aug 29 2022 *)
%Y Eighth column (m=7) of triangle A112705.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 31 2005
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