OFFSET
0,3
COMMENTS
That is, the number of sequences of length n, taking values in {1,...,n} that have nondecreasing first differences (nonnegative second differences).
FORMULA
See Mathematica code.
EXAMPLE
a(3)=16: the 16 sequences are 111, 112, 113, 123, 211, 212, 213, 222, 223, 311, 312, 313, 321, 322, 323 and 333.
MATHEMATICA
(*P[n, k]=number of ways to partition n into exactly k parts*) P[n_Integer, n_Integer] = 1; P[n_Integer, k_Integer] := P[n, k] = Sum[P[n - k, r], {r, 1, Min[n - k, k]}]
(*q[n, k]=number of ways to partition n into k-or-fewer parts*) q[0, 0] = 1; q[n_Integer, 0] = 0; q[n_Integer, k_Integer] := q[n, k] = q[n, k - 1] + P[n, k]
a[n_] := Sum[(n - Max[f, r])*P[r, s]*q[f, n - 1 - s], {r, 0, n - 1}, {s, 0, n - 1}, {f, 0, n - 1}]
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Jacob A. Siehler, Feb 04 2008, Feb 06 2008
STATUS
approved