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A082148
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a(0)=1; for n>=1, a(n) = sum(k=0..n, 10^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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7
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1, 1, 11, 131, 1661, 22101, 305151, 4335711, 63009881, 932449961, 14004694451, 212944033051, 3271618296661, 50711564152381, 792088104593511, 12454801769554551, 196991734871121201, 3131967533789345361
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OFFSET
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0,3
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COMMENTS
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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 10^C(n+1,2). - Philippe DELEHAM, Oct 29 2007
a(n) = upper left term in M^n, M = the production matrix:
1, 1
10, 10, 10
1, 1, 1, 1
10, 10, 10, 10, 10
1, 1, 1, 1, 1, 1
...
- Gary W. Adamson, Jul 08 2011
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REFERENCES
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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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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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G.f. (1+9*x-sqrt(81*x^2-22*x+1))/(20*x).
a(n) = Sum_{k=0..n} A088617(n, k)*10^k*(-9)^(n-k) . - Philippe Deléham, Jan 21 2004
a(n) = (11(2n-1)a(n-1) - 81(n-2)a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - Philippe DELEHAM, Aug 19 2005
a(n) ~ sqrt(20+11*sqrt(10))*(11+2*sqrt(10))^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
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MAPLE
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A082148_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]+10*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A082148_list(17); # Peter Luschny, May 19 2011
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MATHEMATICA
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Table[SeriesCoefficient[(1+9*x-Sqrt[81*x^2-22*x+1])/(20*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
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PROG
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(PARI) a(n)=if(n<1, 1, sum(k=0, n, 10^k/n*binomial(n, k)*binomial(n, k+1)))
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CROSSREFS
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Cf. A001003, A007564, A059231.
Sequence in context: A076357 A015606 A077417 * A075509 A061113 A101334
Adjacent sequences: A082145 A082146 A082147 * A082149 A082150 A082151
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre, May 10 2003
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STATUS
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approved
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