A242499 Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 1.

1, 0, 1, 3, 1, 9, 11, 18, 51, 65, 151, 290, 477, 1043, 1835, 3486, 6931, 12540, 24607, 46797, 87979, 171072, 323269, 619245, 1190619, 2264925, 4357211, 8343322, 15973309, 30711853, 58846191, 113027716, 217192103, 416964202, 801880039, 1541412015, 2963997227

Offset: 1

Author

Alois P. Heinz, May 16 2014

Keywords

• nonn

Comments

With offset 2 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -1.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Column k=1 of A242498.

Programs

Maple

a:= proc(n) option remember;
      `if`(n<6, [0, 1, 0, 1, 3, 1][n+1],
      ((3*n-2)*a(n-2) +(4*n+2)*a(n-3) -(3*n-10)*a(n-4)
       -(4*n-22)*a(n-5) +(n-6)*a(n-6))/(n+2))
    end:
seq(a(n), n=1..50);

Mathematica

a[n_] := a[n] = If[n<6, {0, 1, 0, 1, 3, 1}[[n+1]], ((3n-2)a[n-2] + (4n+2)a[n-3] - (3n-10)a[n-4] - (4n-22)a[n-5] + (n-6)a[n-6])/(n+2)];
Array[a, 50] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

Formula

Recurrence (for n>=5): (n+2)*(16*n^4 - 128*n^3 + 344*n^2 - 352*n + 89)*a(n) = -32*(n+1)*(2*n-5)*a(n-1) + 2*(16*n^5 - 112*n^4 + 264*n^3 - 320*n^2 + 301*n - 89)*a(n-2) + 2*(2*n-5)*(16*n^4 - 80*n^3 + 80*n^2 + 36*n - 53)*a(n-3) - (n-4)*(16*n^4 - 64*n^3 + 56*n^2 + 16*n - 31)*a(n-4). - Vaclav Kotesovec, May 20 2014