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
Keywords
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)
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