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A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1). 3
1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 27, 8, 5, 3, 2, 1, 1, 47, 13, 7, 5, 3, 2, 1, 1, 79, 21, 11, 7, 5, 3, 2, 1, 1, 130, 33, 16, 11, 7, 5, 3, 2, 1, 1, 209, 52, 24, 15, 11, 7, 5, 3, 2, 1, 1, 330, 80, 35, 22, 15, 11, 7, 5, 3, 2, 1, 1, 512, 122, 52, 31, 22, 15, 11, 7, 5, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Weakly unimodal means nondecreasing then nonincreasing.

LINKS

Table of n, a(n) for n=1..88.

FORMULA

T(n, k) = S(n, k) - S(n-k, k) + Sum_j[T(n-k, j)] for j >= k, where S(n, k) = A008284(n, k) = Sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n] = 1.

EXAMPLE

Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.

As an upper right triangle:

  1,  1,  2,  4,  8, 15, ...,

      1,  1,  2,  3,  5,  8, ...,

          1,  1,  2,  3,  5,  7, ...,

              ...

As a left downward triangle, it starts:

   1;

   1, 1;

   2, 1, 1;

   4, 2, 1, 1;

   8, 3, 2, 1, 1;

  15, 5, 3, 2, 1, 1;

  27, 8, 5, 3, 2, 1, 1;

  ...

T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.

CROSSREFS

Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.

Sequence in context: A140998 A048004 A114394 * A140997 A140996 A141020

Adjacent sequences:  A059620 A059621 A059622 * A059624 A059625 A059626

KEYWORD

nonn,tabl

AUTHOR

Henry Bottomley, Feb 01 2001

STATUS

approved

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Last modified August 25 21:00 EDT 2019. Contains 326324 sequences. (Running on oeis4.)