A393918 Triangle read by rows: T(n, k) is the number of compositions of n into parts at least k.

1, 2, 1, 4, 1, 1, 8, 2, 1, 1, 16, 3, 1, 1, 1, 32, 5, 2, 1, 1, 1, 64, 8, 3, 1, 1, 1, 1, 128, 13, 4, 2, 1, 1, 1, 1, 256, 21, 6, 3, 1, 1, 1, 1, 1, 512, 34, 9, 4, 2, 1, 1, 1, 1, 1, 1024, 55, 13, 5, 3, 1, 1, 1, 1, 1, 1, 2048, 89, 19, 7, 4, 2, 1, 1, 1, 1, 1, 1

Offset

1,2

Links

Table of n, a(n) for n=1..78.

Formula

G.f. for column k: (1-x)/(1-x-x^k).

T(n, k) = A099238(n-k, k-1).

T(n, k) = A141539(n-2*k+1, k-1) for k <= (n+1)/2.

Example

T(6, 2) = 5 because the compositions of 6 into parts at least 2 are: 2+2+2, 2+4, 4+2, 3+3, 6.
Triangle begins:
  1;
  2, 1;
  4, 1, 1;
  8, 2, 1, 1;
  16, 3, 1, 1, 1;
  32, 5, 2, 1, 1, 1;
  64, 8, 3, 1, 1, 1, 1;
  ...

Prog

(PARI) T(n, k) = {my(x='x+O('x^(n+1))); polcoef((1-x)/(1-x-x^k), n)}

Crossrefs

Columns 1..15: A011782, A212804, A078012, A017898, A017899, A017900, A017901, A017902, A017903, A017904, A017905, A017906, A017907, A017908, A017909.

Cf. A097805, A099238, A126198, A141539.

Sequence in context: A283826 A088443 A117352 * A137710 A068009 A140168

Adjacent sequences: A393914 A393915 A393917 * A393919 A393920 A393921

Keyword

nonn,tabl

Author

Jason Yuen, May 10 2026

Status

approved