A078009 a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^k*N(n,k) where N(n,k) = C(n,k)*C(n,k+1)/n are the Narayana numbers (A001263).
1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
- Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+4*x - Sqrt(16*x^2-12*x+1))/(10*x) )); // G. C. Greubel, Jun 28 2019 -
Magma
[1] cat [&+[5^k*Binomial(n,k)*Binomial(n,k+1)/n:k in [0..n]]:n in [1..20]]; // Marius A. Burtea, Jan 21 2020
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Maple
A078009_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1; for w from 1 to n do a[w] := a[w-1]+5*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a, list) end: A078009_list(20); # Peter Luschny, May 19 2011
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Mathematica
Table[SeriesCoefficient[(1+4*x-Sqrt[16*x^2-12*x+1])/(10*x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 13 2012 *) a[n_] := Hypergeometric2F1[1 - n, -n, 2, 5]; Table[a[n], {n, 0, 30}] (* Peter Luschny, Mar 19 2018 *) -
PARI
a(n)=sum(k=0,n,5^k/n*binomial(n,k)*binomial(n,k+1))
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Sage
a=((1+4*x -sqrt(16*x^2-12*x+1))/(10*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 28 2019
Formula
G.f.: (1 + 4*x - sqrt(16*x^2 - 12*x + 1))/(10*x).
a(n) = Sum_{k=0..n} A088617(n, k)*5^k*(-4)^(n-k). - Philippe Deléham, Jan 21 2004
With offset 1 : a(1)=1, a(n) = -4*a(n-1) + 5*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
a(n+1) = Sum_{k=0..floor(n/2)} C(n, 2*k)*C(k)*6^(n-2k)*5^k; - Paul Barry, May 19 2005
a(n) = ( 6*(2*n-1)*a(n-1) - 16*(n-2)*a(n-2) ) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
1, 1
5, 5, 5
1, 1, 1, 1
5, 5, 5, 5, 5
1, 1, 1, 1, 1, 1
... (End)
a(n) ~ sqrt(10+6*sqrt(5))*(6+2*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012. Equivalently, a(n) ~ 2^(2*n) * phi^(2*n + 1) / (5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
a(n) = A127848(n) for n > 0. - Philippe Deléham, Apr 03 2013
G.f.: 1/(1 - x/(1 - 5*x/(1 - x/(1 - 5*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017
a(n) = hypergeom([1 - n, -n], [2], 5). - Peter Luschny, Mar 19 2018
Comments