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A082147
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a(0)=1, for n>=1 a(n)=sum(k=0,n,8^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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6
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1, 1, 9, 89, 945, 10577, 123129, 1476841, 18130401, 226739489, 2878666857, 37006326777, 480750990993, 6301611631473, 83240669582937, 1106980509493641, 14808497812637121, 199138509770855489, 2690461489090104009
(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+4)*x+1))/((2*m+2)*x) are given by : a(n)=sum(k=0,n,(m+1)^k*N(n,k))
The Hankel transform of this sequence is 8^C(n+1,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2007
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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+7*x-sqrt(49*x^2-18*x+1))/(16*x)
a(n) = Sum_{k=0..n} A088617(n, k)*8^k*(-7)^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 21 2004
a(n) = [9(2n-1)a(n-1) - 49(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) = upper left term in M^n, M = the production matrix:
1, 1
8, 8, 8
1, 1, 1, 1
8, 8, 8, 8, 8
1, 1, 1, 1, 1, 1
...
- Gary W. Adamson, Jul 08 2011
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MAPLE
| A082147_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]+8*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A082147_list(18); # Peter Luschny, May 19 2011
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PROG
| (PARI) a(n)=if(n<1, 1, sum(k=0, n, 8^k/n*binomial(n, k)*binomial(n, k+1)))
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CROSSREFS
| Cf. A001003, A007564, A059231.
Sequence in context: A109002 A142991 A152267 * A095722 A199759 A069573
Adjacent sequences: A082144 A082145 A082146 * A082148 A082149 A082150
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003
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