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A278987
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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.
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4
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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
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OFFSET
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1,12
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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,
...
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MAPLE
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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:
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CROSSREFS
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The words for b=9 are listed in A273978.
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KEYWORD
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AUTHOR
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STATUS
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approved
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