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).

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, 56, 421, 469, 78, 77, 689, 1013, 264, 5, 101, 1113, 2059, 786, 37, 135, 1750, 4021, 2097, 189, 176, 2712, 7558, 5179, 751, 8, 231, 4128, 13780, 11998, 2558, 73, 297, 6208, 24440, 26400, 7762, 429

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Keywords

• nonn
• tabf
• look

Comments

Same as A238343, with zeros omitted.

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

Row sums are A011782.

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

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

Examples

			Triangle starts:
00:    1;
01:    1;
02:    2;
03:    3,     1;
04:    5,     3;
05:    7,     9;
06:   11,    19,      2;
07:   15,    41,      8;
08:   22,    77,     29;
09:   30,   142,     81,      3;
10:   42,   247,    205,     18;
11:   56,   421,    469,     78;
12:   77,   689,   1013,    264,      5;
13:  101,  1113,   2059,    786,     37;
14:  135,  1750,   4021,   2097,    189;
15:  176,  2712,   7558,   5179,    751,     8;
16:  231,  4128,  13780,  11998,   2558,    73;
17:  297,  6208,  24440,  26400,   7762,   429;
18:  385,  9201,  42358,  55593,  21577,  1945,  13;
19:  490, 13502,  71867, 112814,  55867,  7465, 139;
20:  627, 19585, 119715, 221639, 136478, 25317, 927;
...
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..250, flattened

Crossrefs

Cf. A045883.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
       add(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[b[n-j, j]*If[jJean-François Alcover, Feb 11 2015, after Maple *)

Formula

Sum_{k=0..floor(n/3)} k * T(n,k) = A045883(n-2) for n>=2.