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 A241719 Number T(n,k) of compositions of n into distinct parts with exactly k descents; triangle T(n,k), n>=0, 0<=k<=max(floor((sqrt(1+8*n)-3)/2),0), read by rows. 12
 1, 1, 1, 2, 1, 2, 1, 3, 2, 4, 6, 1, 5, 7, 1, 6, 11, 2, 8, 16, 3, 10, 31, 15, 1, 12, 36, 16, 1, 15, 55, 29, 2, 18, 71, 41, 3, 22, 101, 65, 5, 27, 147, 144, 32, 1, 32, 188, 179, 35, 1, 38, 245, 269, 63, 2, 46, 327, 382, 93, 3, 54, 421, 549, 148, 5, 64, 540, 739, 205, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Rows n = 0..500, flattened EXAMPLE T(6,0) = 4: [6], [2,4], [1,5], [1,2,3]. T(6,1) = 6: [5,1], [4,2], [3,1,2], [1,3,2], [2,1,3], [2,3,1]. T(6,2) = 1: [3,2,1]. T(7,0) = 5: [7], [3,4], [2,5], [1,6], [1,2,4]. T(7,1) = 7: [6,1], [4,3], [5,2], [2,1,4], [1,4,2], [2,4,1], [4,1,2]. T(7,2) = 1: [4,2,1]. Triangle T(n,k) begins: 00:   1; 01:   1; 02:   1; 03:   2,   1; 04:   2,   1; 05:   3,   2; 06:   4,   6,   1; 07:   5,   7,   1; 08:   6,  11,   2; 09:   8,  16,   3; 10:  10,  31,  15,  1; 11:  12,  36,  16,  1; 12:  15,  55,  29,  2; 13:  18,  71,  41,  3; 14:  22, 101,  65,  5; 15:  27, 147, 144, 32, 1; MAPLE g:= proc(u, o) option remember; `if`(u+o=0, 1, expand(       add(g(u+j-1, o-j)  , j=1..o)+       add(g(u-j, o+j-1)*x, j=1..u)))     end: b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;       `if`(n>m, 0, `if`(n=m, x^i,       expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))     end: T:= n-> (p-> (q-> seq(coeff(q, x, i), i=0..degree(q)))(add(          coeff(p, x, k)*g(0, k), k=0..degree(p))))(b(n\$2)): seq(T(n), n=0..20); MATHEMATICA g[u_, o_] := g[u, o] = If[u+o == 0, 1, Expand[Sum[g[u+j-1, o-j], {j, 1, o}] + Sum[g[u-j, o+j-1]*x, {j, 1, u}]]]; b[n_, i_] := b[n, i] = Module[{m}, m = i*(i+1)/2; If[n>m, 0, If[n == m, x^i, Expand[b[n, i-1] + If[i>n, 0, x*b[n-i, i-1]]]]]]; T[n_] := Function [p, Function[q, Table[Coefficient[q, x, i], {i, 0, Exponent[q, x]}]][Sum[Coefficient[p, x, k]*g[0, k], {k, 0, Exponent[p, x]}]]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Apr 28 2014, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000009, A241720, A241721, A241722, A241723, A241724, A241725, A241726, A241727, A241728, A241729. Row sums give A032020. T(A000217(k+1)-1,k-1) = A000041(k) for k>0. Cf. A052146. Sequence in context: A161282 A226517 A185214 * A269572 A029198 A029175 Adjacent sequences:  A241716 A241717 A241718 * A241720 A241721 A241722 KEYWORD nonn,tabf,look AUTHOR Alois P. Heinz, Apr 27 2014 STATUS approved

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Last modified February 27 08:13 EST 2020. Contains 332300 sequences. (Running on oeis4.)