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A176287 Diagonal sums of number triangle A092392.

%I A176287 #28 Oct 13 2024 17:41:53
%S A176287 1,2,7,23,81,291,1066,3955,14818,55937,212428,810664,3106167,11942261,
%T A176287 46047897,178000950,689580319,2676598447,10406929687,40525045518,
%U A176287 158022343991,616950024334,2411395005316,9434753907065,36948692202031
%N A176287 Diagonal sums of number triangle A092392.
%C A176287 Hankel transform is A176288.
%H A176287 G. C. Greubel, <a href="/A176287/b176287.txt">Table of n, a(n) for n = 0..1000</a>
%F A176287 G.f.: 1/(sqrt(1-4*x)*(1-x^2*c(x))) = 2/(sqrt(1-4*x)*(2-x+x*sqrt(1-4*x))), c(x) the g.f. of A000108.
%F A176287 a(n) = Sum_{k=0..floor(n/2)} C(2n-3k,n-k).
%F A176287 a(n) = Sum_{k=0..n} A000984(k)*A132364(n-k).
%F A176287 D-finite with recurrence: 2*n*a(n) +(6-11*n)*a(n-1) +(13*n-16)*a(n-2) +2*(5-n)*a(n-3) +3*(2-3*n)*a(n-4) +2*(2*n-5)*a(n-5)=0. - _R. J. Mathar_, Nov 15 2012 [Verified with Maple's FindRE and MinimalRecurrence functions, _Georg Fischer_, Nov 03 2022]
%F A176287 a(n) ~ 2^(2*n+3) / (7*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 04 2014
%p A176287 seq( add(binomial(2*n-3*k, n-k), k=0..floor(n/2)) , n=0..25); # _G. C. Greubel_, Nov 25 2019
%t A176287 CoefficientList[Series[2/(Sqrt[1-4*x]*(2-x+x*Sqrt[1-4*x])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 04 2014 *)
%t A176287 a[n_]:= Sum[Binomial[2*n-3*k, n-k], {k, 0, Floor[n/2]}]; Table[a[n], {n,0,25}] (* _G. C. Greubel_, Oct 19 2016 *)
%o A176287 (PARI) a(n) = sum(k=0, n\2, binomial(2*n-3*k, n-k)); \\ _Michel Marcus_, Oct 20 2016
%o A176287 (Magma) [ &+[Binomial(2*n-3*k, n-k): k in [0..Floor(n/2)]] : n in [0..25]]; // _G. C. Greubel_, Nov 25 2019
%o A176287 (Sage) [sum(binomial(2*n-3*k, n-k) for k in (0..floor(n/2))) for n in (0..25)] # _G. C. Greubel_, Nov 25 2019
%o A176287 (GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Binomial(2*n-3*k, n-k) )); # _G. C. Greubel_, Nov 25 2019
%Y A176287 Cf. A000108, A000984, A092392, A132364.
%K A176287 easy,nonn
%O A176287 0,2
%A A176287 _Paul Barry_, Apr 14 2010