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

1, 3, 10, 34, 118, 417, 1497, 5448, 20063, 74649, 280252, 1060439, 4040413, 15488981, 59701236, 231236830, 899559100, 3513314664, 13770811198, 54152480421, 213585706927, 844723104691, 3349274471386, 13310603555085, 53012829376985, 211560158583657, 845856494229348, 3387782725245302, 13590698721293800, 54604853170818121, 219706932640295523

Offset: 0

Author

Paul Barry, Dec 26 2010

Keywords

• nonn

Comments

Hankel transform is the (1,-1) Somos-4 sequence A178079.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A025254.

Programs

Magma

m:=50; R:=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

Mathematica

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 *)
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 *)

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)));
vector(22,n,a(n-1))

Formula

a(n) = A025254(n+2).

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).

From Vaclav Kotesovec, Mar 02 2014: (Start)

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).

G.f.: (1 - 3*x - x^2 - sqrt(x^4 + 2*x^3 + 7*x^2 - 6*x + 1))/(2*x^3).

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.

(End)