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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

Table of n, a(n) for n=1..78.

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 A128760 A057884

Adjacent sequences:  A278984 A278985 A278986 * A278988 A278989 A278990

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and N. J. A. Sloane, Dec 06 2016

STATUS

approved

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Last modified October 23 11:46 EDT 2021. Contains 348212 sequences. (Running on oeis4.)