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A178578 Diagonal sums of second binomial transform of the Narayana triangle A001263.

%I A178578 #37 Feb 09 2025 06:34:54
%S A178578 1,3,10,34,118,417,1497,5448,20063,74649,280252,1060439,4040413,
%T A178578 15488981,59701236,231236830,899559100,3513314664,13770811198,
%U A178578 54152480421,213585706927,844723104691,3349274471386,13310603555085,53012829376985,211560158583657,845856494229348,3387782725245302,13590698721293800,54604853170818121,219706932640295523
%N A178578 Diagonal sums of second binomial transform of the Narayana triangle A001263.
%C A178578 Hankel transform is the (1,-1) Somos-4 sequence A178079.
%H A178578 G. C. Greubel, <a href="/A178578/b178578.txt">Table of n, a(n) for n = 0..1000</a>
%F A178578 a(n) = A025254(n+2).
%F A178578 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j)*C(j,k)*C(j+1,k)*2^(n-k-j)/(k+1).
%F A178578 From _Vaclav Kotesovec_, Mar 02 2014: (Start)
%F A178578 Recurrence: (n+3)*a(n) = 3*(2*n+3)*a(n-1) - 7*n*a(n-2) - (2*n-3)*a(n-3) - (n-3)*a(n-4).
%F A178578 G.f.: (1 - 3*x - x^2 - sqrt(x^4 + 2*x^3 + 7*x^2 - 6*x + 1))/(2*x^3).
%F A178578 a(n) ~ (130-216*r-64*r^2-29*r^3) * sqrt(2*r^3+14*r^2-18*r+4) / (4 * sqrt(Pi) * n^(3/2) * r^n), where r = 1/6*(-3 + sqrt(3*(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))) - sqrt(-66 - 3*(1009 - 24*sqrt(183))^(1/3) - 3*(1009 + 24*sqrt(183))^(1/3) + 216*sqrt(3/(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))))) = 0.23742047190096998... is the root of the equation r^4 + 2*r^3 + 7*r^2 - 6*r + 1 = 0.
%F A178578 (End)
%t A178578 Table[Sum[Sum[Binomial[n-k,j]*Binomial[j,k]*Binomial[j+1,k]*2^(n-k-j)/(k+1),{j,0,n-k}],{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 02 2014 *)
%t A178578 CoefficientList[Series[(1-3*x-x^2 -Sqrt[x^4+2*x^3+7*x^2-6*x+1])/(2*x^3), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 14 2018 *)
%o A178578 (PARI) a(n)=sum(k=0,floor(n/2), sum(j=0,n-k,binomial(n-k,j)*binomial(j,k)*binomial(j+1,k)*2^(n-k-j)/(k+1)));
%o A178578 vector(22,n,a(n-1))
%o A178578 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 -3*x-x^2 - Sqrt(x^4+2*x^3+7*x^2-6*x+1))/(2*x^3))); // _G. C. Greubel_, Aug 14 2018
%Y A178578 Cf. A025254.
%K A178578 nonn
%O A178578 0,2
%A A178578 _Paul Barry_, Dec 26 2010