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A275080
Triangle g(n,m) by rows: the number of m-compositions of Carlitz type of n without zero rows.
0
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
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
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Jul 15 2016
STATUS
approved