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A278987 Array read by antidiagonals downwards: T(b,n) = number of words of length n over an alphabet of size b that are in standard order and which have the property that every letter that appears in the word is repeated. 4
0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 0, 1, 11, 4, 1, 1, 0, 1, 26, 11, 4, 1, 1, 0, 1, 57, 41, 11, 4, 1, 1, 0, 1, 120, 162, 41, 11, 4, 1, 1, 0, 1, 247, 610, 162, 41, 11, 4, 1, 1, 0, 1, 502, 2165, 715, 162, 41, 11, 4, 1, 1, 0, 1, 1013, 7327, 3425, 715, 162, 41, 11, 4, 1, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,12
COMMENTS
We study words made of letters from an alphabet of size b, where b >= 1. We assume the letters are labeled {1,2,3,...,b}. There are b^n possible words of length n.
We say that a word is in "standard order" if it has the property that whenever a letter i appears, the letter i-1 has already appeared in the word. This implies that all words begin with the letter 1.
LINKS
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
FORMULA
The number of words of length n over an alphabet of size b that are in standard order and in which every symbol that appears in a word is repeated is Sum_{j = 1..b} A008299(n,j).
EXAMPLE
The array begins:
0,.1,..1,...1,...1,...1,...1,....1..; b=1,
0,.1,..4,...8,..16,..32,..64,..128..; b=2,
0,.1,..4,..14,..41,.122,.365,.1094..; b=3,
0,.1,..4,..14,..51,.187,.715,.2795..; b=4,
0,.1,..4,..14,..51,.202,.855,.3845..; b=5,
0,.1,..4,..14,..51,.202,.876,.4111..; b=6,
...
Rows b=1 through b=4 of the array are A000012, A000295 (or A130103), A278988, A278989.
MAPLE
with(combinat);
A008299 := proc(n, k) local i, j, t1;
if k<1 or k>floor(n/2) then t1:=0; else
t1 := add( (-1)^i*binomial(n, i)*add( (-1)^j*(k - i - j)^(n - i)/(j!*(k - i - j)!), j = 0..k - i), i = 0..k); fi; t1; end;
f3:=proc(L, b) global A008299; local i; add(A008299(L, i), i=1..b); end;
Q3:=b->[seq(f3(L, b), L=1..40)];
for b from 1 to 6 do lprint(Q3(b)); od:
CROSSREFS
The words for b=9 are listed in A273978.
Sequence in context: A117414 A085639 A158972 * A135302 A364026 A128760
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and N. J. A. Sloane, Dec 06 2016
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)