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A078009
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a(0)=1, for n>=1 a(n)=sum(k=0,n,5^k*N(n,k)) where N(n,k) =1/n*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
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9
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1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| More generally coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+2)*x+1))/(2*m*x) are given by : a(n)=sum(k=0,n,(m+1)^k*N(n,k))
a(n) is the series reversion of x(1-5x)/(1-4x); a(n+1) is the series reversion of x/(1+6x+5x^2); a(n+1) counts (6,5)-Motzkin paths of length n, where there are 6 colors available for the H(1,0) steps and 5 for the U(1,1) steps. - Paul Barry (pbarry(AT)wit.ie), May 19 2005
The Hankel transform of this sequence is 5^C(n+1,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2007
a(n) is the number of Schroder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 4 colors. Example: a(2)=6 because we have UDUD, UUDD, UBD, UGD, URD, and UYD, where U=(1,1), D=(1,-1), while B, G, R, and Y are, respectively, blue, green, red, and yellow (2,0)-steps. - Emeric Deutsch, May 02 2011
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REFERENCES
| Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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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) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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 (benoit7848c(AT)orange.fr), Mar 16 2004
a(n+1)=sum{k=0..floor(n/2), C(n, 2k)C(k)6^(n-2k)*5^k}; - Paul Barry (pbarry(AT)wit.ie), May 19 2005
a(n) = [6(2n-1)a(n-1) - 16(n-2)a(n-2)] / (n+1) for n>=2, a(0) = a(1) = 1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 19 2005
a(n)=if(n=0,1,if(n=1,1,6*((2n-1)/(n+1))*a(n-1)-16*((n-2)/(n+1))*a(n-2))). [From Paul Barry (pbarry(AT)wit.ie), Oct 22 2009]
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
...
- Gary W. Adamson, Jul 08 2011
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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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PROG
| (PARI) a(n)=sum(k=0, n, 5^k/n*binomial(n, k)*binomial(n, k+1))
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CROSSREFS
| Cf. A001003, A007564, A059231.
Sequence in context: A152107 A143023 * A127848 A113573 A083161 A077147
Adjacent sequences: A078006 A078007 A078008 * A078010 A078011 A078012
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003
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