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A293741
Number of sets of nonempty words with a total of n letters over binary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
5
1, 1, 2, 5, 10, 23, 51, 111, 243, 530, 1156, 2497, 5421, 11662, 25179, 53991, 116035, 248025, 531045, 1131943, 2415495, 5135914, 10927905, 23182313, 49199819, 104154950, 220543471, 465997148, 984704560, 2076988713, 4380764650, 9225209928, 19424814305
OFFSET
0,3
LINKS
FORMULA
G.f.: Product_{j>=1} (1+x^j)^A001405(j).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
b(n-i*j, i-1)*binomial(binomial(i, floor(i/2)), j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* Binomial[Binomial[i, Floor[i/2]], j], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 29 2019, after Alois P. Heinz *)
PROG
(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, i): return 1 if n==0 else 0 if i<1 else sum([b(n - i*j, i - 1)*binomial(binomial(i, i//2), j) for j in range(n//i + 1)])
def a(n): return b(n, n)
print([a(n) for n in range(36)]) # Indranil Ghosh, Oct 15 2017
CROSSREFS
Column k=2 of A293112.
Cf. A001405.
Sequence in context: A087640 A116953 A099516 * A291559 A297074 A099963
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 15 2017
STATUS
approved