This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A112701 #23 Mar 31 2025 18:07:27
%S A112701 1,8,106,1821,35435,741329,16270997,369570944,8613236374,204812473608,
%T A112701 4949266755812,121188396669810,3000342229924222,74979188061284522,
%U A112701 1888846103011564082,47915719069874907917,1222954711282739097587
%N A112701 Partial sum of Catalan numbers (A000108) multiplied by powers of 7.
%H A112701 Robert Israel, <a href="/A112701/b112701.txt">Table of n, a(n) for n = 0..693</a>
%F A112701 a(n) = Sum_{k=0..n} A000108(k)*7^k.
%F A112701 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 A112701 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 A112701 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
%F A112701 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
%p A112701 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 A112701 map(f, [$0..50]); # _Robert Israel_, Aug 04 2020
%t A112701 CatalanNumber[#]*7^#& /@ Range[0, 20] // Accumulate (* _Jean-François Alcover_, Aug 29 2022 *)
%Y A112701 Column m=7 of triangle A112705.
%Y A112701 Partial sums of A156266.
%Y A112701 Cf. A000108, A000420.
%K A112701 nonn,easy
%O A112701 0,2
%A A112701 _Wolfdieter Lang_, Oct 31 2005