A192070 -id:A192070 - cp's OEIS frontend

A192250 0-sequence of reduction of central binomial coefficient sequence by x^2 -> x+1.

1, 1, 7, 27, 167, 923, 5543, 32999, 200309, 1221329, 7503033, 46301793, 286971677, 1784658077, 11131825877, 69611130917, 436270168817, 2739539507957, 17232530582057, 108564692241257, 684901029237677, 4326215549824277, 27357682806703397

Offset: 1

Clark Kimberling, Jun 27 2011

nonn

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

Cf. A192232, A192251, A192070.

c[n_] := (2 n)!/(n! n!); (* central binomial coefficients, A000984 *)
Table[c[n], {n, 0, 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,
   30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192250 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192251 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]  (* A192070 *)
(* by Peter J. C. Moses, Jun 20 2011 *)

Conjecture: (n-1)*(n-2)*a(n) -(5*n-7)*(n-2)*a(n-1) -2*(2*n-3)*(3*n-8)*a(n-2) +4*(2*n-3)*(2*n-5)*a(n-3)=0. - R. J. Mathar, May 04 2014

A192251 1-sequence of reduction of central binomial coefficient sequence by x^2 -> x+1.

Original entry on oeis.org

0, 2, 8, 48, 258, 1518, 8910, 53526, 323796, 1976876, 12138456, 74921904, 464320368, 2887660168, 18011618368, 112633305568, 705899650498, 4432668783838, 27882818399038, 175661366346838, 1108193133814138, 6999963827434378, 44265660573879298

Offset: 1

Clark Kimberling, Jun 27 2011

nonn

See A192232, A192250.

Cf. A192232, A192250, A192070.

[&+[Fibonacci(k)*Binomial(2*k,k): k in [0..n]]: n in [0..28]]; // Vincenzo Librandi, Feb 04 2016

(See A192250.)
Table[Sum[Fibonacci[i] Binomial[2 i, i], {i, 0, n - 1}], {n, 23}] (* Michael De Vlieger, Feb 01 2016 *)

a(n) = 2*A192070(n).
Conjecture: (n-1)*(n-2)*a(n) -(5*n-7)*(n-2)*a(n-1) -2*(2*n-3)*(3*n-8)*a(n-2) +4*(2*n-3)*(2*n-5)*a(n-3)=0. - R. J. Mathar, May 04 2014
a(n) = Sum_{i=0..n-1} A000045(i)*binomial(2*i, i). - John M. Campbell, Feb 01 2016

cp's OEIS Frontend

A192250 0-sequence of reduction of central binomial coefficient sequence by x^2 -> x+1.

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