cp's OEIS Frontend

A165409 Transform of 2^n by the aerated Catalan triangle A165408.

%I A165409 #20 Feb 05 2025 05:49:45
%S A165409 1,2,4,10,24,56,136,328,784,1896,4576,11008,26592,64192,154752,373696,
%T A165409 902144,2176640,5255424,12687488,30621952,73931392,178484736,
%U A165409 430845952,1040176640,2511199232,6062209024,14635617280,35333443584,85300015104
%N A165409 Transform of 2^n by the aerated Catalan triangle A165408.
%C A165409 Hankel transform is A165410.
%H A165409 Vincenzo Librandi, <a href="/A165409/b165409.txt">Table of n, a(n) for n = 0..200</a>
%F A165409 G.f.: 1/(1-2*x-2*x^3*c(2*x^3)) = 2/(1-4*x+sqrt(1-8*x^3)) = (1-4*x-sqrt(1-8*x^3) )/(4*x*(1-2*x-x^2)), c(x) the g.f. of A000108.
%F A165409 G.f.: 1/(1-2*x-2*x^3/(1-2*x^3/(1-2*x^3/(1-2*x^3/(1-... (continued fraction).
%F A165409 a(n) = Sum_{k=ceiling(n/3)..n} 2^k*C((n+k)/2, k)*((3*k-n)/2 + 1)*(1+(-1)^(n-k))/(2*(k+1)).
%F A165409 a(n) = Sum_{k=0..n} 2^k * A165408(n,k).
%F A165409 a(n) = Sum_{k=0..n+1} Pell(n-k+1)*(0^k - 2^((k-2)/2)*A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).
%F A165409 (n+1)*a(n) = 2(n+1)*a(n-1) + (n+1)*a(n-2) + 4*(2*n-7)*a(n-3) - 8(2*n-7)*a(n-4) - 4*(2*n-7)*a(n-5). - _R. J. Mathar_, Nov 17 2011
%F A165409 a(n) ~ (4+sqrt(2)) * (1+sqrt(2))^n / 8. - _Vaclav Kotesovec_, Feb 01 2014
%t A165409 CoefficientList[Series[2/(1-4*x+Sqrt[1-8*x^3]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)
%o A165409 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2/(1-4*x+Sqrt(1-8*x^3)) )); // _G. C. Greubel_, Nov 10 2022
%o A165409 (SageMath)
%o A165409 def A165408(n,k): return 0 if (n>3*k) else (1+(-1)^(n-k))*(3*k-n+2)*binomial(int((n+k)/2), k)/(4*(k+1))
%o A165409 def A165409(n): return sum(2^k*A165408(n,k) for k in range(n+1))
%o A165409 [A165409(n) for n in range(41)] # _G. C. Greubel_, Nov 10 2022
%Y A165409 Cf. A000108, A000129, A165408, A165410.
%K A165409 easy,nonn
%O A165409 0,2
%A A165409 _Paul Barry_, Sep 17 2009