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 A340973 #32 Aug 19 2025 09:33:40
%S A340973 1,7,67,721,8179,95557,1137709,13725439,167204947,2052215893,
%T A340973 25338173497,314356676179,3915672171229,48938691421627,
%U A340973 613404577267843,7707619156442401,97058716523798227,1224551690144551237
%N A340973 Generating function Sum_{n >= 0} a(n)*x^n = 1/sqrt((1-x)*(1-13*x)).
%H A340973 Seiichi Manyama, <a href="/A340973/b340973.txt">Table of n, a(n) for n = 0..899</a>
%F A340973 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(2*k,k).
%F A340973 a(n) = [x^n] (1+7*x+9*x^2)^n.
%F A340973 n * a(n) = 7 * (2*n-1) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1.
%F A340973 E.g.f.: exp(7*x) * BesselI(0,6*x). - _Ilya Gutkovskiy_, Feb 01 2021
%F A340973 a(n) ~ 13^(n + 1/2) / (2 * sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Nov 13 2021
%F A340973 From _Seiichi Manyama_, Aug 19 2025: (Start)
%F A340973 a(n) = (1/4)^n * Sum_{k=0..n} 13^k * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A340973 a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(n,k) * binomial(2*k,k). (End)
%t A340973 a[n_] := Sum[3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Feb 01 2021 *)
%t A340973 nxt[{n_,a_,b_}]:={n+1,b,(7*b(2n+1)-13*n*a)/(n+1)}; Join[{1},NestList[nxt,{2,7,67},20] [[All,2]]] (* _Harvey P. Dale_, Apr 27 2022 *)
%o A340973 (PARI) my(N=20, x='x+O('x^N)); Vec(1/sqrt((1-x)*(1-13*x)))
%o A340973 (PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(2*k, k));
%o A340973 (PARI) a(n) = polcoef((1+7*x+9*x^2)^n, n);
%Y A340973 Column k=3 of A340970.
%Y A340973 Cf. A322246, A337167.
%K A340973 nonn
%O A340973 0,2
%A A340973 _Seiichi Manyama_, Feb 01 2021