A192069 -id:A192069

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A192248

0-sequence of reduction of binomial coefficient sequence B(n,4)=A000332 by x^2 -> x+1.

+10
3

1, 1, 16, 51, 191, 569, 1619, 4259, 10694, 25709, 59743, 134818, 296798, 639518, 1352498, 2813750, 5769200, 11676395, 23358450, 46239770, 90667076, 176244326, 339887026, 650715076, 1237467151, 2338753519, 4394813644, 8214444389

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

Table of n, a(n) for n=1..28.

FORMULA

Conjecture: G.f.: -x*(-1+5*x-20*x^2+30*x^3-25*x^4+8*x^5) / ( (x-1)*(x^2+x-1)^5 ). - R. J. Mathar, May 04 2014

MATHEMATICA

c[n_] := n (n + 1) (n + 2) (n + 3)/24; (* binomial B(n, 4), A000332 *)

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 + 1]

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,

40}]

Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192248 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192249 *)

Table[Coefficient[Part[t, n]/5, x, 1], {n, 1, 40}] (* A192069 *)

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

CROSSREFS

Cf. A192232, A192249, A192069.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 27 2011

STATUS

approved

A192249

1-sequence of reduction of binomial coefficient sequence B(n,4)=A000332 by x^2 -> x+1.

+10
2

0, 5, 20, 90, 300, 930, 2610, 6900, 17295, 41605, 96660, 218145, 480225, 1034765, 2188385, 4552745, 9334760, 18892805, 37794765, 74817520, 146702410, 285169310, 549948760, 1052879110, 2002263910, 3784182685, 7110957850, 13291250220