A238341 - OEIS

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

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OFFSET

0,8

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.

LINKS

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

EXAMPLE

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;

...

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 *)

CROSSREFS

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

Sequence in context: A081658 A187253 A022904 * A242451 A363978 A262964

Adjacent sequences: A238338 A238339 A238340 * A238342 A238343 A238344

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 25 2014

STATUS

approved