A278987 - OEIS

%I #18 Dec 06 2016 20:44:07

%S 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,

%T 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,

%U 2165,715,162,41,11,4,1,1,0,1,1013,7327,3425,715,162,41,11,4,1,1,0

%N 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.

%C 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.

%C 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.

%H Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>

%F 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).

%e The array begins:

%e 0,.1,..1,...1,...1,...1,...1,....1..; b=1,

%e 0,.1,..4,...8,..16,..32,..64,..128..; b=2,

%e 0,.1,..4,..14,..41,.122,.365,.1094..; b=3,

%e 0,.1,..4,..14,..51,.187,.715,.2795..; b=4,

%e 0,.1,..4,..14,..51,.202,.855,.3845..; b=5,

%e 0,.1,..4,..14,..51,.202,.876,.4111..; b=6,

%e ...

%e Rows b=1 through b=4 of the array are A000012, A000295 (or A130103), A278988, A278989.

%p with(combinat);

%p A008299 := proc(n,k) local i,j,t1;

%p if k<1 or k>floor(n/2) then t1:=0; else

%p 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;

%p f3:=proc(L,b) global A008299; local i; add(A008299(L,i),i=1..b); end;

%p Q3:=b->[seq(f3(L,b),L=1..40)];

%p for b from 1 to 6 do lprint(Q3(b)); od:

%Y The words for b=9 are listed in A273978.

%K nonn,tabl

%O 1,12

%A _Joerg Arndt_ and _N. J. A. Sloane_, Dec 06 2016