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Diagonal sums of number triangle A114709.
2

%I #19 Sep 03 2022 21:06:00

%S 1,0,3,6,31,126,589,2772,13485,66780,336207,1714698,8841627,46015002,

%T 241394073,1275137448,6776728825,36208438488,194388488155,

%U 1048061471886,5672504958327,30808982057046,167864115588325,917271225518076,5025659929354981

%N Diagonal sums of number triangle A114709.

%H Vincenzo Librandi, <a href="/A186940/b186940.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. (for offset 1): (1+3*x-2*x^2-sqrt(1-6*x+x^2))/(2*(3+x-3*x^2+x^3)).

%F Conjecture: 3*(n+1)*a(n) +(10-17*n)*a(n-1) -6*(n+1)*a(n-2) +10*(2*n-1)*a(n-3) +9*(1-n)*a(n-4) +(n-2)*a(n-5) =0. - _R. J. Mathar_, Nov 17 2011

%F a(n) ~ sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^(n+2) / (36 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2014

%t Rest[CoefficientList[Series[(1+3*x-2*x^2-Sqrt[1-6*x+x^2])/(2*(3+x-3*x^2+x^3)), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Feb 01 2014 *)

%Y Cf. A114709.

%Y Hankel transform is A186941.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Mar 01 2011

%E More terms from _Vincenzo Librandi_, Feb 14 2014