A192252 0-sequence of reduction of (n!) by x^2 -> x+1.

1, 1, 3, 9, 57, 417, 4017, 44337, 568497, 8188977, 131568177, 2326992177, 44958134577, 941649129777, 21254190979377, 514247427715377, 13277149259395377, 364340640790147377, 10588931448837763377, 324919870905259651377, 10496883167091791491377

Offset: 0

Author

Clark Kimberling, Jun 27 2011

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

Cf. A192232, A192253.

Programs

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

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

Conjecture: a(n) = Sum_{k=0..n} A052554(k). - Sean A. Irvine, Jul 14 2022