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A192749 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
0, 1, 6, 16, 35, 68, 124, 217, 370, 620, 1027, 1688, 2760, 4497, 7310, 11864, 19235, 31164, 50468, 81705, 132250, 214036, 346371, 560496, 906960, 1467553, 2374614, 3842272, 6216995, 10059380, 16276492, 26335993, 42612610, 68948732, 111561475 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..34.

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

Cf. A192744, A192232.

Sequence in context: A266677 A083053 A083046 * A160997 A199629 A098943

Adjacent sequences:  A192746 A192747 A192748 * A192750 A192751 A192752

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 09 2011

STATUS

approved

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Last modified October 22 09:57 EDT 2018. Contains 316433 sequences. (Running on oeis4.)