|
| |
|
|
A074358
|
|
Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = b*nu(n-1) + lambda*(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda)=(2,2).
|
|
6
|
|
|
|
0, 0, 0, 4, 20, 80, 288, 976, 3184, 10112, 31488, 96576, 292672, 878336, 2614784, 7731456, 22728448, 66482176, 193617920, 561718272, 1624101888, 4681535488, 13457924096, 38592008192, 110419341312, 315287830528, 898583560192
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: 4*x^3*(x + 1)/(2*x^2 + 2*x - 1)^2 (conjectured). - Chai Wah Wu, May 30 2016
a(n) = (1/18)*((1 + sqrt(3))^n*(-9 + 2*sqrt(3)) - (1 - sqrt(3))^n*(9 + 2*sqrt(3)) + 3*((1 - sqrt(3))^n + (1 + sqrt(3))^n)*n) for n > 0 (conjectured). - Colin Barker, Nov 17 2017
a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n > 4 (conjectured). - Colin Barker, Nov 17 2017
|
|
|
EXAMPLE
|
The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3) = 16 + 4q, nu(4) = 44 + 20q + 12q^2, nu(5) = 120 + 80q + 64q^2 + 40q^3 + 8q^4, so the coefficients of q^1 are 0,0,0,4,20,80.
|
|
|
MAPLE
|
nu := proc(b, lambda, n) global q; local qp, i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b, lambda, n-1)+lambda*qp*nu(b, lambda, n-2)) ; end: A074358 := proc(n) RETURN( coeftayl(nu(2, 2, n), q=0, 1) ) ; end: for n from 0 to 30 do printf("%d, ", A074358(n)) ; od ; # R. J. Mathar, Sep 20 2006
|
|
|
MATHEMATICA
|
nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n-1] + 2*Total[q^Range[0, n-2] ]*nu[n-2] // Expand;
a[n_] := Coefficient[nu[n], q, 1];
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
