A321959 a(n) = [x^n] ((1 - x)*x)/((1 - 2*x)^2*(2*x^2 - 2*x + 1)).

0, 1, 5, 16, 42, 100, 228, 512, 1144, 2544, 5616, 12288, 26656, 57408, 122944, 262144, 556928, 1179392, 2490112, 5242880, 11010560, 23069696, 48235520, 100663296, 209713152, 436203520, 905965568, 1879048192, 3892322304, 8053080064, 16643014656, 34359738368

Offset

0,3

Links

Table of n, a(n) for n=0..31.

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Formula

a(n) = Sum_{k=0..n} A323100(n - k, k).

a(n) = n! [x^n] exp(x)*(exp(x)*(2*x + 1) - sin(x) - cos(x))/2.

a(n) = 2*((2*n+2)*a(n-3) - (3*n+2)*a(n-2) + (2*n+1)*a(n-1))/n for n >= 4.

a(2^n - 1) = 2^(2^n + n - 2) if n>1. - Michael Somos, Sep 30 2022

Example

G.f. = x + 5*x^2 + 16*x^3 + 42*x^4 + 100*x^5 + 228*x^6 + ... - Michael Somos, Sep 30 2022

Maple

ogf := ((1 - x)*x)/((1 - 2*x)^2*(2*x^2 - 2*x + 1));
ser := series(ogf, x, 32): seq(coeff(ser, x, n), n=0..31);

Mathematica

LinearRecurrence[{6, -14, 16, -8}, {0, 1, 5, 16}, 32] (* Georg Fischer, May 08 2021 *)

Prog

(PARI) {a(n) = if(n<0, 0, polcoeff( x*(1 - x) / ((1 - 2*x)^2*(1 - 2*x + 2*x^2)), n))}; /* Michael Somos, Sep 30 2022 */

Crossrefs

Antidiagonal sums of A323100.

Sequence in context: A055796 A002662 A143962 * A066634 A241794 A034358

Adjacent sequences: A321956 A321957 A321958 * A321960 A321961 A321962

Keyword

nonn,easy

Author

Peter Luschny, Jan 12 2019

Status

approved