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 A085364 #27 Aug 22 2025 11:02:00
%S A085364 1,6,60,654,7458,87378,1042152,12587730,153479508,1885010946,
%T A085364 23285957604,289018502682,3601315495050,45023019250398,
%U A085364 564465885846216,7094214579174558,89351097367355826,1127492973620753010
%N A085364 a(0)=1, for n>0: a(n) = 6*13^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
%C A085364 From _G. C. Greubel_, May 23 2020: (Start)
%C A085364 This sequence is part of a class of sequences, for m >= 0, with the properties:
%C A085364 a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k).
%C A085364 a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k).
%C A085364 n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
%C A085364 a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
%C A085364 (4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j).
%C A085364 G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ).
%C A085364 This sequence is the case of m=3. (End)
%H A085364 Vincenzo Librandi, <a href="/A085364/b085364.txt">Table of n, a(n) for n = 0..200</a>
%F A085364 G.f.: sqrt((1-x)/(1-13*x))
%F A085364 Sum_{i=0..n} Sum_{j=0..i} a(j)*a(i-j) = 13^n.
%F A085364 D-finite with recurrence: n*a(n) = 2*(7*n-4)*a(n-1) - 13*(n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012
%F A085364 a(n) ~ 2*sqrt(3)*13^(n-1/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 14 2012
%F A085364 a(0) = 1; a(n) = (6/n) * Sum_{k=0..n-1} (n+k) * a(k). - _Seiichi Manyama_, Mar 28 2023
%F A085364 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A085364 a(n) = (1/4)^n * Sum_{k=0..n} 13^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A085364 a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%p A085364 seq(coeff(series( sqrt((1-x)/(1-13*x)) , x, n+1), x, n), n = 0..30); # _G. C. Greubel_, May 23 2020
%t A085364 CoefficientList[Series[Sqrt[(1-x)/(1-13x)], {x, 0, 25}], x]
%o A085364 (PARI) my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-13*x))) \\ _Joerg Arndt_, May 10 2013
%o A085364 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-13*x)) )); // _G. C. Greubel_, May 23 2020
%o A085364 (Sage)
%o A085364 def A085362_list(prec):
%o A085364 P.<x> = PowerSeriesRing(ZZ, prec)
%o A085364 return P( sqrt((1-x)/(1-13*x)) ).list()
%o A085364 A085362_list(30) # _G. C. Greubel_, May 23 2020
%Y A085364 Cf. A001022 (13^n), A085362, A085363.
%K A085364 easy,nonn,changed
%O A085364 0,2
%A A085364 Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003