A099929 Central Pellonomial coefficients.

1, 2, 30, 2436, 1166438, 3248730940, 52755584809356, 4992850354675749192, 2754130291777980970686150, 8854642279944231931659815098860, 165923943638796574201560736475319416580, 18121679707218614746613513717704194807763644600

Offset: 0

Author

Ralf Stephan, Nov 03 2004

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A000129, A099927, A256831.

Programs

Maple

p:= proc(n) p(n):= `if`(n<2, n, 2*p(n-1)+p(n-2)) end:
f:= proc(n) f(n):= `if`(n=0, 1, p(n)*f(n-1)) end:
a:= n-> f(2*n)/f(n)^2:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 15 2013

Mathematica

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,{n,0,20}] (* Vaclav Kotesovec, Apr 10 2015 *)

Sage

P=[lucas_number1(n, 2, -1) for n in [0..30]]
[prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]] # Tom Edgar, Apr 10 2015

Sage

def a(n): return ((1+sqrt(2))^n^2*q_binomial(2*n, n, -(3-2*sqrt(2)))).simplify_full() # Seiichi Manyama, May 10 2025

Formula

a(n) = A099927(2n, n).

a(n) ~ (1+sqrt(2))^(n^2) / c, where c = A256831 = 1.141982569667791206028... . - Vaclav Kotesovec, Apr 10 2015

a(n) = (1 + sqrt(2))^(n^2) * q-binomial(2*n, n, -(sqrt(2) - 1)^2). - Seiichi Manyama, May 10 2025