A192069 -id:A192069 - cp's OEIS frontend

A192248 0-sequence of reduction of binomial coefficient sequence B(n,4)=A000332 by x^2 -> x+1.

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

Offset: 1

Clark Kimberling, Jun 27 2011

nonn

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

Cf. A192232, A192249, A192069.

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 *)

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

A192249 1-sequence of reduction of binomial coefficient sequence B(n,4)=A000332 by x^2 -> x+1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 27 2011

nonn

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

Cf. A192232, A192248, A192069.

Mathematica
```
(See A192248.)
```

a(n) = 5*A192069(n).

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

cp's OEIS Frontend

A192248 0-sequence of reduction of binomial coefficient sequence B(n,4)=A000332 by x^2 -> x+1.

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