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Irregular triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=floor(n/3).
14

%I #20 Feb 11 2015 04:32:15

%S 1,1,2,3,1,5,3,7,9,11,19,2,15,41,8,22,77,29,30,142,81,3,42,247,205,18,

%T 56,421,469,78,77,689,1013,264,5,101,1113,2059,786,37,135,1750,4021,

%U 2097,189,176,2712,7558,5179,751,8,231,4128,13780,11998,2558,73,297,6208,24440,26400,7762,429

%N Irregular triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=floor(n/3).

%C Same as A238343, with zeros omitted.

%C Columns k=0-10 give: A000041, A241626, A241627, A241628, A241629, A241630, A241631, A241632, A241633, A241634, A241635.

%C Row sums are A011782.

%C T(3n,n) = A000045(n+1).

%C T(3n+1,n) = A136376(n+1).

%H Joerg Arndt and Alois P. Heinz, <a href="/A238344/b238344.txt">Rows n = 0..250, flattened</a>

%e Triangle starts:

%e 00: 1;

%e 01: 1;

%e 02: 2;

%e 03: 3, 1;

%e 04: 5, 3;

%e 05: 7, 9;

%e 06: 11, 19, 2;

%e 07: 15, 41, 8;

%e 08: 22, 77, 29;

%e 09: 30, 142, 81, 3;

%e 10: 42, 247, 205, 18;

%e 11: 56, 421, 469, 78;

%e 12: 77, 689, 1013, 264, 5;

%e 13: 101, 1113, 2059, 786, 37;

%e 14: 135, 1750, 4021, 2097, 189;

%e 15: 176, 2712, 7558, 5179, 751, 8;

%e 16: 231, 4128, 13780, 11998, 2558, 73;

%e 17: 297, 6208, 24440, 26400, 7762, 429;

%e 18: 385, 9201, 42358, 55593, 21577, 1945, 13;

%e 19: 490, 13502, 71867, 112814, 55867, 7465, 139;

%e 20: 627, 19585, 119715, 221639, 136478, 25317, 927;

%e ...

%p b:= proc(n, i) option remember; `if`(n=0, 1, expand(

%p add(b(n-j, j)*`if`(j<i, x, 1), j=1..n)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):

%p seq(T(n), n=0..20);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, j]*If[j<i, x, 1], {j, 1, n} ]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Feb 11 2015, after Maple *)

%K nonn,tabf,look

%O 0,3

%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014