A192760 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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

Links

Table of n, a(n) for n=0..35.

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

Cf. A192744, A192232.

Sequence in context: A301101 A266340 A266339 * A295964 A292765 A357282

Adjacent sequences: A192757 A192758 A192759 * A192761 A192762 A192763

Keyword

nonn,easy

Author

Clark Kimberling, Jul 09 2011

Status

approved