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A189845 Number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=3+max(prefix) for k>=1. 4
1, 1, 4, 22, 150, 1200, 10922, 110844, 1236326, 14990380, 195895202, 2740062260, 40789039078, 643118787708, 10696195808162, 186993601880756, 3425688601198118, 65586903427253532, 1309155642001921026, 27185548811026532692, 586164185027289760806 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..481 (first 67 terms from Vincenzo Librandi)
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.4, pp. 364-366
FORMULA
E.g.f. of sequence starting 1,4,22,.. is exp(x+exp(x)+exp(2*x)/2+exp(3*x)/3-11/6) = exp(x+sum(j=1,3, (exp(j*x)-1)/j)) = 1+4*x+11*x^2+25*x^3+50*x^4+5461/60*x^5 +...
EXAMPLE
For n=0 there is one empty string; for n=1 there is one string [0]; for n=2 there are 4 strings [00], [01], [02], and [03];
for n=3 there are a(3)=22 strings:
01: [ 0 0 0 ],
02: [ 0 0 1 ],
03: [ 0 0 2 ],
04: [ 0 0 3 ],
05: [ 0 1 0 ],
06: [ 0 1 1 ],
07: [ 0 1 2 ],
08: [ 0 1 3 ],
09: [ 0 1 4 ],
10: [ 0 2 0 ],
11: [ 0 2 1 ],
12: [ 0 2 2 ],
13: [ 0 2 3 ],
14: [ 0 2 4 ],
15: [ 0 2 5 ],
16: [ 0 3 0 ],
17: [ 0 3 1 ],
18: [ 0 3 2 ],
19: [ 0 3 3 ],
20: [ 0 3 4 ],
21: [ 0 3 5 ],
22: [ 0 3 6 ].
MAPLE
b:= proc(n, m) option remember; `if`(n=0, 1,
add(b(n-1, max(m, j)), j=1..m+3))
end:
a:= n-> b(n, -2):
seq(a(n), n=0..25); # Alois P. Heinz, Jun 15 2018
MATHEMATICA
b[n_, m_] := b[n, m] = If[n==0, 1, Sum[b[n-1, Max[m, j]], {j, 1, m+3}]];
a[n_] := b[n, -2];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 03 2020, after Alois P. Heinz *)
PROG
(PARI) x='x+O('x^66);
egf=exp(x+sum(j=1, 3, (exp(j*x)-1)/j)); /* (off by one!) */
concat([1], Vec(serlaplace(egf)))
CROSSREFS
Column k=3 of A305962.
Sequence in context: A346764 A228883 A307439 * A039304 A349022 A267219
KEYWORD
nonn
AUTHOR
Joerg Arndt, Apr 29 2011
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)