A238345 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the first occurrence of a largest part, n>=1, 1<=k<=n.

1, 2, 0, 3, 1, 0, 5, 2, 1, 0, 8, 5, 2, 1, 0, 14, 9, 6, 2, 1, 0, 24, 18, 12, 7, 2, 1, 0, 43, 33, 25, 16, 8, 2, 1, 0, 77, 62, 49, 35, 21, 9, 2, 1, 0, 140, 115, 95, 73, 49, 27, 10, 2, 1, 0, 256, 215, 181, 148, 108, 68, 34, 11, 2, 1, 0, 472, 401, 346, 291, 230, 158, 93, 42, 12, 2, 1, 0, 874, 753, 657, 569, 470, 353, 228, 125, 51, 13, 2, 1, 0

Offset: 1

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Keywords

• nonn
• tabl

Comments

Column k=1: T(n,1) = A079500(n) = A007059(n+1).

Row sums are A011782.

Examples

			Triangle starts:
01:     1;
02:     2,    0;
03:     3,    1,    0;
04:     5,    2,    1,    0;
05:     8,    5,    2,    1,    0;
06:    14,    9,    6,    2,    1,    0;
07:    24,   18,   12,    7,    2,    1,    0;
08:    43,   33,   25,   16,    8,    2,    1,   0;
09:    77,   62,   49,   35,   21,    9,    2,   1,   0;
10:   140,  115,   95,   73,   49,   27,   10,   2,   1,   0;
11:   256,  215,  181,  148,  108,   68,   34,  11,   2,   1,  0;
12:   472,  401,  346,  291,  230,  158,   93,  42,  12,   2,  1,  0;
13:   874,  753,  657,  569,  470,  353,  228, 125,  51,  13,  2,  1, 0;
14:  1628, 1416, 1250, 1102,  943,  753,  533, 324, 165,  61, 14,  2, 1, 0;
15:  3045, 2673, 2380, 2126, 1866, 1558, 1188, 791, 453, 214, 72, 15, 2, 1, 0;
...
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened

Programs

Maple

g:= proc(n, m) option remember; `if`(n=0, 1,
       add(g(n-j, min(n-j, m)), j=1..min(n, m)))
    end:
h:= proc(n, t, m) option remember; `if`(n=0, 0,
      `if`(t=1, add(g(n-j, j), j=m+1..n),
       add(h(n-j, t-1, max(m, j)), j=1..n)))
    end:
T:= (n, k)-> h(n, k, 0):
seq(seq(T(n, k), k=1..n), n=1..15);

Mathematica

g[n_, m_] := g[n, m] = If[n == 0, 1, Sum[g[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]; h[n_, t_, m_] := h[n, t, m] = If[n == 0, 0, If[t == 1, Sum[g[n-j, j], {j, m+1, n}], Sum[h[n-j, t-1, Max[m, j]], {j, 1, n}]]]; T[n_, k_] := h[n, k, 0]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)