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 A241701 Number T(n,k) of Carlitz compositions of n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows. 14
 1, 1, 1, 2, 1, 2, 2, 3, 4, 4, 8, 2, 5, 13, 5, 6, 21, 12, 8, 33, 27, 3, 10, 50, 53, 11, 12, 73, 98, 31, 15, 106, 174, 78, 5, 18, 150, 296, 175, 22, 22, 209, 486, 363, 72, 27, 289, 781, 715, 204, 8, 32, 393, 1222, 1342, 510, 43, 38, 529, 1874, 2421, 1168, 159 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS No two adjacent parts of a Carlitz composition are equal. LINKS Alois P. Heinz, Rows n = 0..250, flattened FORMULA Sum_{k=0..floor(n/3)} (k+1) * T(n,k) = A285994(n) (for n>0). EXAMPLE T(6,0) = 4: [6], [1,5], [2,4], [1,2,3]. T(6,1) = 8: [4,2], [5,1], [3,1,2], [1,3,2], [1,4,1], [2,3,1], [2,1,3], [1,2,1,2]. T(6,2) = 2: [3,2,1], [2,1,2,1]. T(7,0) = 5: [7], [3,4], [1,6], [2,5], [1,2,4]. T(7,1) = 13: [4,3], [6,1], [5,2], [2,1,4], [4,1,2], [1,4,2], [2,3,2], [3,1,3], [1,5,1], [2,4,1], [1,2,3,1], [1,3,1,2], [1,2,1,3]. T(7,2) = 5: [4,2,1], [2,1,3,1], [3,1,2,1], [1,3,2,1], [1,2,1,2,1]. Triangle T(n,k) begins: 00:   1; 01:   1; 02:   1; 03:   2,   1; 04:   2,   2; 05:   3,   4; 06:   4,   8,   2; 07:   5,  13,   5; 08:   6,  21,  12; 09:   8,  33,  27,   3; 10:  10,  50,  53,  11; 11:  12,  73,  98,  31; 12:  15, 106, 174,  78,   5; 13:  18, 150, 296, 175,  22; 14:  22, 209, 486, 363,  72; 15:  27, 289, 781, 715, 204, 8; MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, expand(       add(`if`(j=i, 0, b(n-j, j)*`if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)): seq(T(n), n=0..20); MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[If[j == i, 0, b[n-j, j]*If[j0. Cf. A238344, A285994. Sequence in context: A244788 A078660 A239239 * A060177 A238212 A255723 Adjacent sequences:  A241698 A241699 A241700 * A241702 A241703 A241704 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Apr 27 2014 STATUS approved

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Last modified January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)