cp's OEIS Frontend

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.

%I A278987 #18 Dec 06 2016 20:44:07
%S A278987 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 A278987 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 A278987 2165,715,162,41,11,4,1,1,0,1,1013,7327,3425,715,162,41,11,4,1,1,0
%N 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.
%C A278987 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 A278987 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 A278987 Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%F A278987 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 A278987 The array begins:
%e A278987 0,.1,..1,...1,...1,...1,...1,....1..; b=1,
%e A278987 0,.1,..4,...8,..16,..32,..64,..128..; b=2,
%e A278987 0,.1,..4,..14,..41,.122,.365,.1094..; b=3,
%e A278987 0,.1,..4,..14,..51,.187,.715,.2795..; b=4,
%e A278987 0,.1,..4,..14,..51,.202,.855,.3845..; b=5,
%e A278987 0,.1,..4,..14,..51,.202,.876,.4111..; b=6,
%e A278987 ...
%e A278987 Rows b=1 through b=4 of the array are A000012, A000295 (or A130103), A278988, A278989.
%p A278987 with(combinat);
%p A278987 A008299 := proc(n,k) local i,j,t1;
%p A278987 if k<1 or k>floor(n/2) then t1:=0; else
%p A278987 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 A278987 f3:=proc(L,b) global A008299; local i; add(A008299(L,i),i=1..b); end;
%p A278987 Q3:=b->[seq(f3(L,b),L=1..40)];
%p A278987 for b from 1 to 6 do lprint(Q3(b)); od:
%Y A278987 The words for b=9 are listed in A273978.
%K A278987 nonn,tabl
%O A278987 1,12
%A A278987 _Joerg Arndt_ and _N. J. A. Sloane_, Dec 06 2016