A340973 - OEIS

%I #25 Apr 27 2022 16:29:09

%S 1,7,67,721,8179,95557,1137709,13725439,167204947,2052215893,

%T 25338173497,314356676179,3915672171229,48938691421627,

%U 613404577267843,7707619156442401,97058716523798227,1224551690144551237

%N Generating function Sum_{n >= 0} a(n)*x^n = 1/sqrt((1-x)*(1-13*x)).

%H Seiichi Manyama, <a href="/A340973/b340973.txt">Table of n, a(n) for n = 0..899</a>

%F a(n) = Sum_{k=0..n} 3^k*binomial(n,k)*binomial(2*k,k).

%F a(n) = [x^n] (1+7*x+9*x^2)^n.

%F n * a(n) = 7 * (2*n-1) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1.

%F E.g.f.: exp(7*x) * BesselI(0,6*x). - _Ilya Gutkovskiy_, Feb 01 2021

%F a(n) ~ 13^(n + 1/2) / (2 * sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Nov 13 2021

%t 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 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 (PARI) my(N=20, x='x+O('x^N)); Vec(1/sqrt((1-x)*(1-13*x)))

%o (PARI) {a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(2*k, k))}

%o (PARI) {a(n) = polcoef((1+7*x+9*x^2)^n, n)}

%Y Column k=3 of A340970.

%Y Cf. A322246, A337167.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 01 2021