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A192252 0-sequence of reduction of (n!) by x^2 -> x+1. 2
1, 1, 3, 9, 57, 417, 4017, 44337, 568497, 8188977, 131568177, 2326992177, 44958134577, 941649129777, 21254190979377, 514247427715377, 13277149259395377, 364340640790147377, 10588931448837763377, 324919870905259651377, 10496883167091791491377 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

After the tenth term, the final digit is 7, for terms in both A192252 and A192253. After the 100th term, the final 6 digits of each term of A192252 are 9,3,1,3,7,7.

LINKS

Table of n, a(n) for n=0..20.

FORMULA

Conjecture: a(n) +(-n-1)*a(n-1) -n*(n-2)*a(n-2) +n*(n-1)*a(n-3)=0. - R. J. Mathar, May 04 2014

EXAMPLE

The sequence (n!)=(1,1,2,6,24,120,...) provides coefficients for the power series 1+x+2x^2+6x^3+..., of which the (n+1)st partial sum is the polynomial p(x)=1+x+2x^2+...+(n!)x^n, of which reduction by x^2 -> x+1 (as presented at A192232) is A192252(n)+x*A192253(n).

MATHEMATICA

c[n_] := n!; (* A000142 *)

Table[c[n], {n, 1, 15}]

q[x_] := x + 1;

p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n]

reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};

t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 50}]

Table[Coefficient[Part[t, n], x, 0], {n, 1, 50}]  (* A192252 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 50}]  (* A192253 *)

Table[Coefficient[(-7 + Part[t, n])/10, x, 0], {n, 1, 30}]

(* by Peter J. C. Moses, Jun 20 2011 *)

CROSSREFS

Cf. A192232, A192253.

Sequence in context: A292333 A294785 A040175 * A105466 A261244 A018504

Adjacent sequences:  A192249 A192250 A192251 * A192253 A192254 A192255

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 27 2011

STATUS

approved

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Last modified March 28 15:50 EDT 2020. Contains 333089 sequences. (Running on oeis4.)