A268147 A double binomial sum involving absolute values.

0, 16, 512, 12288, 262144, 5242880, 100663296, 1879048192, 34359738368, 618475290624, 10995116277760, 193514046488576, 3377699720527872, 58546795155816448, 1008806316530991104, 17293822569102704640, 295147905179352825856, 5017514388048998039552

Offset: 0

Author

Richard P. Brent, Jan 27 2016

Keywords

• easy
• nonn

Comments

A fast algorithm follows from Theorem 1 of Brent et al. article.

Links

• Colin Barker, Table of n, a(n) for n = 0..800
• Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477v2 [math.CO], 2016.
• Index entries for linear recurrences with constant coefficients, signature (32,-256).

Crossrefs

Cf. A000984, A002894, A166337.

Programs

Maple

a:= proc(n) option remember;
      16*`if`(n<2, n, n*a(n-1)/(n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 29 2016

Mathematica

Table[n*16^n, {n, 0, 20}] (* Jean-François Alcover, Oct 24 2016 *)
LinearRecurrence[{32,-256},{0,16},20] (* Harvey P. Dale, Jul 19 2018 *)

PARI

a(n) = sum(k=-n,n, sum(l=-n,n,binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k-l)^2));

PARI

concat(0, Vec(16*x/(1-16*x)^2 + O(x^20))) \\ Colin Barker, Feb 11 2016

PARI

a(n)=n*16^n \\ Charles R Greathouse IV, May 10 2016

Formula

a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k-l)^2).

From Colin Barker, Feb 11 2016: (Start)

a(n) = n*16^n.

a(n) = 32*a(n-1)-256*a(n-2) for n>1.

G.f.: 16*x / (1-16*x)^2.

(End)