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Third diagonal (M=3) sequence of triangle A113647, called Y(2,1).
3

%I #10 May 22 2013 05:47:14

%S 1,15,113,783,5361,36879,255985,1794063,12689393,90505231,650379249,

%T 4705157135,34244198385,250572963855,1842382110705,13605619630095,

%U 100872203796465,750556607938575,5602962592235505,41952165966643215

%N Third diagonal (M=3) sequence of triangle A113647, called Y(2,1).

%H Vincenzo Librandi, <a href="/A115150/b115150.txt">Table of n, a(n) for n = 0..300</a>

%F a(n)= A113647(n+2, n+1), n>=0.

%F G.f.: ((4*x-2+x^2) + 2*(1-4*x)*c(2*x))/((x^2)*(1+x)), with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers).

%F Recurrence: (n-1)*(n+3)*a(n) = (7*n^2 + 2*n + 3)*a(n-1) + 4*n*(2*n+1)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012

%F a(n) ~ 2^(3*n+9)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012

%t CoefficientList[Series[((4*x-2+x^2)+2*(1-4*x)*(1-Sqrt[1-8*x])/(4*x))/((x^2)*(1+x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)

%Y a(n)=A115138(n+2), n>=0.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jan 13 2006