%I A086885
%S A086885 2,3,7,4,13,34,5,21,73,209,6,31,136,501,1546,7,43,229,1045,4051,13327,
               8,
%T A086885 57,358,1961,9276,37633,130922,9,73,529,3393,19081,93289,394353,1441729,
%U A086885 10,91,748,5509,36046,207775,1047376,4596553,17572114,11,111,1021,8501
%N A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats 
               can be occupied by any number k (0<=k<=j<=i) of persons.
%C A086885 2
%C A086885 3 7
%C A086885 4 13 34
%C A086885 5 21 73 209
%C A086885 6 31 136 501 1546
%C A086885 Compare with A088699. [From Peter Bala (pbala(AT)toucansurf.com), Sep 
               17 2008]
%H A086885 Ed Jones, <a href="http://mathforum.org/discuss/sci.math/t/528848">Number 
               of seatings</a>
%F A086885 a(n)=T(i, j) with n=(i*(i-1))/2+j; T(i, 1)=i+1, T(i, j)=T(i, j-1)+i*T(i-1, 
               j-1) for j>1
%F A086885 The role of seats and persons may be interchanged, so T(i, j)=T(j, i).
%F A086885 T(i, j) = j!*LaguerreL(j, i-j, -1). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Aug 25 2003
%F A086885 T(i, j) = Sum_{k=0..j} k!*binomial(i, k)*binomial(j, k). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Aug 25 2003
%e A086885 One person:
%e A086885 T(1,1)=a(1)=2: 0,1 (seat empty or occupied)
%e A086885 T(2,1)=a(2)=3: 00,10,01 (both seats empty, left seat occupied, right 
               seat occupied)
%e A086885 Two persons:
%e A086885 T(2,2)=a(3)=7: 00,10,01,20,02,12,21
%e A086885 T(3,2)=a(5)=13: 000,100,010,001,200,020,002,120,102,012,210,201,021
%Y A086885 Diagonal: A002720, first subdiagonal: A000262, 2nd subdiagonal: A052852, 
               3rd subdiagonal: A062147, 4th subdiagonal: A062266, 5th subdiagonal: 
               A062192, 2nd row/column: A002061.
%Y A086885 Sequence in context: A143528 A118810 A026259 this_sequence A082734 A021425 
               A060940
%Y A086885 Adjacent sequences: A086882 A086883 A086884 this_sequence A086886 A086887 
               A086888
%K A086885 nonn,tabl
%O A086885 1,1
%A A086885 Hugo Pfoertner (hugo(AT)pfoertner.org), Aug 22 2003