%I #18 Oct 28 2023 09:57:09

%S 1,0,1,0,1,3,0,3,12,13,0,4,45,108,75,0,7,148,672,1056,541,0,14,477,

%T 3622,10028,11520,4683,0,23,1502,18174,79508,155840,140256,47293,0,39,

%U 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

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

%H E. Munarini, M. Poneti, and S. Rinaldi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Rinaldi/rinaldi.html">Matrix compositions</a>, JIS 12 (2009) 09.4.8, Table 4.

%e 1 ;

%e 0 1 ;

%e 0 1 3 ;

%e 0 3 12 13 ;

%e 0 4 45 108 75;

%e 0 7 148 672 1056 541 ;

%e 0 14 477 3622 10028 11520 4683 ;

%e 0 23 1502 18174 79508 155840 140256 47293;

%e 0 39 4678 87474 570521 1705915 2566554 1894032 545835;

%p z := proc(n,m)

%p kmax := n+1 ;

%p add((-1)^k*(1-(1-x^k)^m)/(1-x^k)^m,k=1..kmax) ;

%p 1/(1+%) ;

%p coeftayl(%,x=0,n) ;

%p end proc:

%p g := proc(n,m)

%p add(binomial(m,k)*(-1)^(m-k)*z(n,k),k=0..m) ;

%p end proc:

%p seq(seq(g(n,m),m=0..n),n=0..12) ;

%t 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}]];

%t g[n_, m_] := Sum[Binomial[m, k]*(-1)^(m-k)*z[n, k], {k, 0, m}];

%t Table[Table[g[n, m], {m, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Oct 28 2023, after _R. J. Mathar_'s program *)

%Y Cf. A003242 (column m=1).

%Y Main diagonal gives A000670.

%K nonn,tabl

%O 0,6

%A _R. J. Mathar_, Jul 15 2016