login
A192760
Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
5
0, 1, 4, 9, 18, 33, 58, 99, 166, 275, 452, 739, 1204, 1957, 3176, 5149, 8342, 13509, 21870, 35399, 57290, 92711, 150024, 242759, 392808, 635593, 1028428, 1664049, 2692506, 4356585, 7049122, 11405739, 18454894, 29860667, 48315596, 78176299
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
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
Sequence in context: A301101 A266340 A266339 * A295964 A292765 A357282
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved