A256799 - OEIS

%I #18 Jan 23 2016 21:51:19

%S 1,1,6,203,40222,46410442,312163223724,12237378320283699,

%T 2796071362211148193590,3723566980632561787914135870,

%U 28901575272390972687956930234335380,1307480498356321410289575304307661963042110,344746842780849469098742541704318199701366091840620

%N Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

%C 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).

%H Alois P. Heinz, <a href="/A256799/b256799.txt">Table of n, a(n) for n = 0..50</a>

%H Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

%F 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).

%F a(n) ~ 2^(3/2) * (1+sqrt(2))^(n^2-n-1) / c, where c = A256831 = 1.141982569667791206028... . - _Vaclav Kotesovec_, Apr 10 2015

%e a(5) = Pell(10)..Pell(7)/Pell(5)..Pell(1) = (2378*985*408*169)/(29*12*5*2*1) = 46410442.

%e a(3) = A099927(6,3)/Pell(3) = 2436/12 = 203.

%p p:= n-> (<<2|1>, <1|0>>^n)[1, 2]:

%p a:= n-> mul(p(i), i=n+2..2*n)/mul(p(i), i=1..n):

%p seq(a(n), n=0..12);  # _Alois P. Heinz_, Apr 10 2015

%t 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 *)

%o (Sage)

%o P=[lucas_number1(n, 2, -1) for n in [0..30]]

%o [1/P[n+1]*prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]]

%Y Cf. A000129, A099927, A003150, A099929, A256831, A256832.

%K nonn

%O 0,3

%A _Tom Edgar_, Apr 10 2015