OFFSET
0,3
COMMENTS
The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+n+2 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.
Form an array with m(1,j) = m(j,1) = j for j >= 1 in the top row and left column, and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j). The sum of the terms in the n-th antidiagonal is a(n). - J. M. Bergot, Nov 07 2012
LINKS
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
FORMULA
a(n) = 2*A000045(n+3)-n-4. G.f. x*(-1-x+x^2) / ( (x^2+x-1)*(x-1)^2 ). - R. J. Mathar, Nov 09 2012
a(n) = Sum_{1..n} C(n-i+2,i+1) + C(n-i,i). - Wesley Ivan Hurt, Sep 13 2017
MATHEMATICA
q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 2;
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}] (* A001594 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192760 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved