A241699
Number of Carlitz compositions of n with exactly nine descents.
Original entry on oeis.org
55, 491, 2689, 11608, 42795, 140589, 422681, 1183166, 3121247, 7830626, 18812643, 43515886, 97340687, 211323653, 446587109, 921008294, 1857606992, 3670974168, 7119426996, 13569286673, 25448188412, 47013582854, 85641353036, 153964392218, 273387455379
Offset: 27
-
b:= proc(n, i) option remember; `if`(n=0, 1,
convert(series(add(`if`(i=j, 0, b(n-j, j)*
`if`(j coeff(b(n, 0), x, 9):
seq(a(n), n=27..55);
A241700
Number of Carlitz compositions of n with exactly ten descents.
Original entry on oeis.org
89, 874, 5176, 23882, 93525, 324959, 1029660, 3029025, 8377958, 21992465, 55185174, 133116640, 310082552, 700123670, 1536990597, 3289297465, 6877652389, 14077255217, 28252887221, 55681517610, 107900601488, 205825555915, 386884938126, 717249234951
Offset: 30
-
b:= proc(n, i) option remember; `if`(n=0, 1,
convert(series(add(`if`(i=j, 0, b(n-j, j)*
`if`(j coeff(b(n, 0), x, 10):
seq(a(n), n=30..55);
A285994
Number of increasing runs in all Carlitz compositions of n.
Original entry on oeis.org
0, 1, 1, 4, 6, 11, 26, 46, 84, 167, 313, 576, 1086, 2016, 3710, 6876, 12660, 23196, 42542, 77798, 141910, 258648, 470558, 854644, 1550588, 2809620, 5084588, 9192349, 16601714, 29953754, 53997062, 97257129, 175033355, 314771224, 565664138, 1015841191
Offset: 0
a(1) = 1: (1).
a(2) = 1: (2).
a(3) = 4: (12), (2)(1), (3).
a(4) = 6: (12)(1), (13), (3)(1), (4).
a(5) = 11: (2)(12), (13)(1), (23), (3)(2), (14), (4)(1), (5).
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b:= proc(n, l) option remember; `if`(n=0, [1, 0], add(`if`(j=l, 0,
(p-> p+`if`(j>l, [0, p[1]], 0))(b(n-j, j))), j=1..n))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..40);
-
b[n_, l_] := b[n, l] = If[n == 0, {1, 0}, Sum[If[j == l, {0, 0}, Function[p, p + If[j > l, {0, p[[1]]}, 0]][b[n - j, j]]], {j, 1, n}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2022, after Alois P. Heinz *)
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