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A238344 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).

%I A238344 #22 Apr 30 2025 14:30:39
%S A238344 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 A238344 56,421,469,78,77,689,1013,264,5,101,1113,2059,786,37,135,1750,4021,
%U A238344 2097,189,176,2712,7558,5179,751,8,231,4128,13780,11998,2558,73,297,6208,24440,26400,7762,429
%N A238344 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 A238344 Same as A238343, with zeros omitted.
%C A238344 Columns k=0-10 give: A000041, A241626, A241627, A241628, A241629, A241630, A241631, A241632, A241633, A241634, A241635.
%C A238344 Row sums are A011782.
%C A238344 T(3n,n) = A000045(n+1).
%C A238344 T(3n+1,n) = A136376(n+1).
%H A238344 Joerg Arndt and Alois P. Heinz, <a href="/A238344/b238344.txt">Rows n = 0..250, flattened</a>
%F A238344 Sum_{k=0..floor(n/3)} k * T(n,k) = A045883(n-2) for n>=2.
%e A238344 Triangle starts:
%e A238344 00:    1;
%e A238344 01:    1;
%e A238344 02:    2;
%e A238344 03:    3,     1;
%e A238344 04:    5,     3;
%e A238344 05:    7,     9;
%e A238344 06:   11,    19,      2;
%e A238344 07:   15,    41,      8;
%e A238344 08:   22,    77,     29;
%e A238344 09:   30,   142,     81,      3;
%e A238344 10:   42,   247,    205,     18;
%e A238344 11:   56,   421,    469,     78;
%e A238344 12:   77,   689,   1013,    264,      5;
%e A238344 13:  101,  1113,   2059,    786,     37;
%e A238344 14:  135,  1750,   4021,   2097,    189;
%e A238344 15:  176,  2712,   7558,   5179,    751,     8;
%e A238344 16:  231,  4128,  13780,  11998,   2558,    73;
%e A238344 17:  297,  6208,  24440,  26400,   7762,   429;
%e A238344 18:  385,  9201,  42358,  55593,  21577,  1945,  13;
%e A238344 19:  490, 13502,  71867, 112814,  55867,  7465, 139;
%e A238344 20:  627, 19585, 119715, 221639, 136478, 25317, 927;
%e A238344 ...
%p A238344 b:= proc(n, i) option remember; `if`(n=0, 1, expand(
%p A238344        add(b(n-j, j)*`if`(j<i, x, 1), j=1..n)))
%p A238344     end:
%p A238344 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A238344 seq(T(n), n=0..20);
%t A238344 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 *)
%Y A238344 Cf. A045883.
%K A238344 nonn,tabf,look
%O A238344 0,3
%A A238344 _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014