OFFSET
0,3
COMMENTS
The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+4n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.
a(n+1) is the row sum of row n of the triangle defined by T(n,1)=n*(n-1)+1, T(n,n)=2*n-1, n>=1, and T(r,c)=T(r-1,c)+T(r-2,c-1). The triangle starts 1; 3,3; 7,4,5; 13,7,8,7; 21,14,12,12,9; - J. M. Bergot, Apr 26 2013
LINKS
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
FORMULA
G.f. -x*(1+3*x) / ( (x^2+x-1)*(x-1)^2 ). a(n+1)-a(n) = A053311(n). - R. J. Mathar, Apr 29 2013
MATHEMATICA
q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
(* A053311 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192749 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved