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A256799
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Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).
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1
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1, 1, 6, 203, 40222, 46410442, 312163223724, 12237378320283699, 2796071362211148193590, 3723566980632561787914135870, 28901575272390972687956930234335380, 1307480498356321410289575304307661963042110, 344746842780849469098742541704318199701366091840620
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OFFSET
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0,3
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COMMENTS
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One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from Pell numbers (A000129).
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LINKS
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FORMULA
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a(n) = Pell(2n)Pell(2n-1)...Pell(n+2)/Pell(n)Pell(n-1)...Pell(1) = A099927(2*n,n)/Pell(n+1) = A099929(n)/Pell(n+1), where Pell(k) = A000129(k).
a(n) ~ 2^(3/2) * (1+sqrt(2))^(n^2-n-1) / c, where c = A256831 = 1.141982569667791206028... . - Vaclav Kotesovec, Apr 10 2015
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EXAMPLE
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a(5) = Pell(10)..Pell(7)/Pell(5)..Pell(1) = (2378*985*408*169)/(29*12*5*2*1) = 46410442.
a(3) = A099927(6,3)/Pell(3) = 2436/12 = 203.
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MAPLE
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p:= n-> (<<2|1>, <1|0>>^n)[1, 2]:
a:= n-> mul(p(i), i=n+2..2*n)/mul(p(i), i=1..n):
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MATHEMATICA
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Pell[m_]:=Expand[((1+Sqrt[2])^m-(1-Sqrt[2])^m)/(2*Sqrt[2])]; Table[Product[Pell[k], {k, 1, 2*n}]/(Product[Pell[k], {k, 1, n}])^2 / Pell[n+1], {n, 0, 15}] (* Vaclav Kotesovec, Apr 10 2015 *)
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PROG
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(Sage)
P=[lucas_number1(n, 2, -1) for n in [0..30]]
[1/P[n+1]*prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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