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A360321 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

%I A360321 #23 Aug 22 2025 11:02:17
%S A360321 1,2,16,130,1070,8902,74724,631902,5376840,45990070,395106656,
%T A360321 3407196982,29477061166,255733684010,2224098916300,19384492018770,
%U A360321 169270624419390,1480625235653670,12970844831940000,113785067475668550,999400688480388570
%N A360321 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
%H A360321 Harvey P. Dale, <a href="/A360321/b360321.txt">Table of n, a(n) for n = 0..1000</a>
%F A360321 G.f.: sqrt( (1-5*x)/(1-9*x) ).
%F A360321 n*a(n) = 2*(7*n-6)*a(n-1) - 45*(n-2)*a(n-2).
%F A360321 Sum_{i=0..n} Sum_{j=0..i} (1/5)^i * a(j) * a(i-j) = (9/5)^n.
%F A360321 a(n) ~ 2 * 3^(2*n-1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 04 2023
%F A360321 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A360321 a(n) = (1/4)^n * Sum_{k=0..n} 9^k * 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A360321 a(n) = Sum_{k=0..n} (-1)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%t A360321 Table[Sum[5^(n-k) Binomial[n-1,n-k]Binomial[2k,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Jun 22 2025 *)
%o A360321 (PARI) a(n) = sum(k=0, n, 5^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
%o A360321 (PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-5*x)/(1-9*x)))
%Y A360321 Cf. A063886, A085362, A360317, A360318, A360319, A360322.
%K A360321 nonn,easy,changed
%O A360321 0,2
%A A360321 _Seiichi Manyama_, Feb 03 2023