A238341 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the largest part, n>=0, 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 1, 0, 1, 0, 12, 3, 0, 0, 1, 0, 23, 7, 1, 0, 0, 1, 0, 46, 13, 4, 0, 0, 0, 1, 0, 91, 25, 10, 1, 0, 0, 0, 1, 0, 183, 46, 21, 5, 0, 0, 0, 0, 1, 0, 367, 89, 39, 15, 1, 0, 0, 0, 0, 1, 0, 737, 175, 70, 35, 6, 0, 0, 0, 0, 0, 1, 0, 1478, 351, 125, 71, 21, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Keywords

• nonn
• tabl

Comments

Columns k=0-10 give: A000007, A097979(n-1) for n>0, A243737, A243738, A243739, A243740, A243741, A243742, A243743, A243744, A243745.

T(n^2,n) gives A243746(n).

Row sums are A011782.

Examples

			Triangle starts:
00:  1;
01:  0,     1;
02:  0,     1,    1;
03:  0,     3,    0,   1;
04:  0,     6,    1,   0,   1;
05:  0,    12,    3,   0,   0,   1;
06:  0,    23,    7,   1,   0,   0,  1;
07:  0,    46,   13,   4,   0,   0,  0, 1;
08:  0,    91,   25,  10,   1,   0,  0, 0, 1;
09:  0,   183,   46,  21,   5,   0,  0, 0, 0, 1;
10:  0,   367,   89,  39,  15,   1,  0, 0, 0, 0, 1;
11:  0,   737,  175,  70,  35,   6,  0, 0, 0, 0, 0, 1;
12:  0,  1478,  351, 125,  71,  21,  1, 0, 0, 0, 0, 0, 1;
13:  0,  2962,  710, 229, 131,  56,  7, 0, 0, 0, 0, 0, 0, 1;
14:  0,  5928, 1443, 435, 230, 126, 28, 1, 0, 0, 0, 0, 0, 0, 1,
15:  0, 11858, 2926, 859, 395, 253, 84, 8, 0, 0, 0, 0, 0, 0, 0, 1;
...
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A026794 (the same for partitions), A238342 (the same for smallest part).

Programs

Mathematica

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_, k_] := Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]; a[0, 0] = 1; a[, 0] = 0; Table[a[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* _Jean-François Alcover, Jan 19 2015, after Maple code in A243737 *)