A275080 Triangle g(n,m) by rows: the number of m-compositions of Carlitz type of n without zero rows.

1, 0, 1, 0, 1, 3, 0, 3, 12, 13, 0, 4, 45, 108, 75, 0, 7, 148, 672, 1056, 541, 0, 14, 477, 3622, 10028, 11520, 4683, 0, 23, 1502, 18174, 79508, 155840, 140256, 47293, 0, 39, 4678, 87474, 570521, 1705915, 2566554, 1894032, 545835, 0, 71, 14508, 410379, 3850376, 16529925, 37084794, 45082170, 28159872, 7087261, 0, 124, 44817, 1894116, 24966124, 148188201, 465922722, 831175513, 845735016, 457657776, 102247563

Offset

0,6

Links

Table of n, a(n) for n=0..65.

E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Table 4.

Example

1 ;
0 1 ;
0 1 3 ;
0 3 12 13 ;
0 4 45 108 75;
0 7 148 672 1056 541 ;
0 14 477 3622 10028 11520 4683 ;
0 23 1502 18174 79508 155840 140256 47293;
0 39 4678 87474 570521 1705915 2566554 1894032 545835;

Maple

z := proc(n, m)
    kmax := n+1 ;
    add((-1)^k*(1-(1-x^k)^m)/(1-x^k)^m, k=1..kmax) ;
    1/(1+%) ;
    coeftayl(%, x=0, n) ;
end proc:
g := proc(n, m)
    add(binomial(m, k)*(-1)^(m-k)*z(n, k), k=0..m) ;
end proc:
seq(seq(g(n, m), m=0..n), n=0..12) ;

Mathematica

z[n_, m_] := Module[{kmax, s}, kmax = n+1; s = Sum[(-1)^k*(1-(1-x^k)^m)/ (1-x^k)^m, {k, 1, kmax}]; SeriesCoefficient[1/(1+s), {x, 0, n}]];
g[n_, m_] := Sum[Binomial[m, k]*(-1)^(m-k)*z[n, k], {k, 0, m}];
Table[Table[g[n, m], {m, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 28 2023, after R. J. Mathar's program *)

Crossrefs

Cf. A003242 (column m=1).

Main diagonal gives A000670.

Sequence in context: A157521 A176005 A211963 * A128252 A230675 A327673

Adjacent sequences: A275077 A275078 A275079 * A275081 A275082 A275083

Keyword

nonn,tabl

Author

R. J. Mathar, Jul 15 2016

Status

approved