A026638 - OEIS

%I #20 Jul 03 2024 01:19:24

%S 1,2,8,26,92,332,1220,4538,17036,64412,244928,935684,3588392,13806704,

%T 53271548,206040506,798600332,3101109164,12062148368,46986821516,

%U 183276382472,715748620424,2798274135368,10951009023716,42895901012792,168167959150232,659793819847040

%N a(n) = A026637(2*n, n).

%H Vincenzo Librandi, <a href="/A026638/b026638.txt">Table of n, a(n) for n = 0..1000</a>

%F From _Vaclav Kotesovec_, Oct 21 2012: (Start)

%F G.f.: (3 - (x+1)*sqrt(1-4*x))/((x+2)*sqrt(1-4*x)).

%F Recurrence: 2*n*a(n) = (7*n-4)*a(n-1) + 2*(2*n-1)*a(n-2).

%F a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)) (End)

%t CoefficientList[Series[1/(2+x)+3/((2+x)*Sqrt[1-4*x])-1,{x,0,20}],x] (* _Vaclav Kotesovec_, Oct 21 2012 *)

%o (PARI) my(x='x+O('x^66)); Vec( 1/(2+x)+3/((2+x)*sqrt(1-4*x))-1 ) \\ _Joerg Arndt_, May 04 2013

%o (Magma) [1] cat [n le 2 select 2^(2*n-1) else ((7*n-4)*Self(n-1) + 2*(2*n-1)*Self(n-2))/(2*n): n in [1..40]]; // _G. C. Greubel_, Jul 01 2024

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A026638

%o     if n<3: return 2^(n*(n+1)/2)

%o     else: return ((7*n-4)*a(n-1) + 2*(2*n-1)*a(n-2))/(2*n)

%o [a(n) for n in range(41)] # _G. C. Greubel_, Jul 01 2024

%Y Cf. A026637, A026639, A026640, A026642, A026643, A026644, A026645.

%Y Cf. A026646, A026647, A026966, A026967, A026968, A026969, A026970.

%K nonn

%O 0,2

%A _Clark Kimberling_