%I A121207
%S A121207 1,1,1,1,1,2,1,1,3,5,1,1,4,9,15,1,1,5,14,31,52,1,1,6,20,54,121,203,
%T A121207 1,1,7,27,85,233,523,877,1,1,8,35,125,400,1101,2469,4140,1,1,9,44,
%U A121207 175,635,2046,5625,12611,21147,1,1,10,54,236,952,3488,11226,30846
%N A121207 Triangle read by rows. The definition is by diagonals. The r-th diagonal 
               from the right, for r >= 0, is given by b(0) = b(1) = 1; b(n+1) = 
               Sum_{k=0..n} binomial(n+2,k+r)*a(k).
%C A121207 Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Dec 12 2006. (Start) 
               Consider the row reversal, which is A124496 with an additional left 
               column (A000110 = Bell numbers). The matrix inverse of this triangle 
               is very simple:
%C A121207 1;
%C A121207 -1, 1;
%C A121207 -1, -1, 1;
%C A121207 -1, -2, -1, 1;
%C A121207 -1, -3, -3, -1, 1;
%C A121207 -1, -4, -6, -4, -1, 1;
%C A121207 -1, -5, -10, -10, -5, -1, 1;
%C A121207 -1, -6, -15, -20, -15, -6, -1, 1;
%C A121207 -1, -7, -21, -35, -35, -21, -7, -1, 1;
%C A121207 -1, -8, -28, -56, -70, -56, -28, -8, -1, 1; ...
%C A121207 This gives the recurrence and explains why the Bell numbers appear. (End)
%C A121207 Triangle A160185 = reversal then deletes right border of 1's. [From Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), May 03 2009]
%e A121207 Triangle begins:
%e A121207 1,
%e A121207 1, 1,
%e A121207 1, 1, 2,
%e A121207 1, 1, 3, 5,
%e A121207 1, 1, 4, 9, 15,
%e A121207 1, 1, 5, 14, 31, 52,
%e A121207 1, 1, 6, 20, 54, 121, 203,
%e A121207 1, 1, 7, 27, 85, 233, 523, 877,
%e A121207 1, 1, 8, 35, 125, 400,1101,2469,4140,
%e A121207 1, 1, 9, 44, 175, 635,2046,5625,12611,21147,
%e A121207 1, 1, 10, 54, 236, 952,3488,11226,30846,69161,115975,
%e A121207 1, 1, 11, 65, 309,1366,5579,20425,65676,180474,404663,678570,
%e A121207 1, 1, 12, 77, 395,1893,8494,34685,126817,407787,1120666,2512769,4213597,
%e A121207 1, 1, 13, 90, 495,2550,12432,55818,227550,831915,2675410,7352471,16485691,
               27644437, etc
%p A121207 (Maple program from R. J. Mathar) Gould := proc(n,d) local k; if n<=1 
               then RETURN(1); else
%p A121207 # This is the Jovovic formula with general index 'd'
%p A121207 # where A040027, A045499 etc. use one explicit integer
%p A121207 # Index n+1 is shifted to n from the original formula.
%p A121207 RETURN(add(binomial(n-1+d,k+d)*Gould(k,d),k=0..n-1));
%p A121207 fi;
%p A121207 end:
%p A121207 # row and col refer to the extrapolated super-table:
%p A121207 for row from 0 to 13 do
%p A121207 # working up to row, not row-1, shows also the Bell numbers
%p A121207 # at the end of each row
%p A121207 for col from 0 to row do
%p A121207 # 'diag' is constant for one of A040027, A045499 etc
%p A121207 diag := row-col;
%p A121207 printf("%4d,",Gould(col,diag));
%p A121207 od;
%p A121207 print();
%p A121207 od;
%Y A121207 Diagonals, reading from the right, are A000110, A040027, A045501, A045499, 
               A045500.
%Y A121207 A124496 is a very similar triangle, obtained by reversing the rows and 
               appending a right-most diagonal which is A000110, the Bell numbers.
%Y A121207 A160185 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 03 2009]
%Y A121207 Sequence in context: A068098 A135722 A049513 this_sequence A097084 A143327 
               A094954
%Y A121207 Adjacent sequences: A121204 A121205 A121206 this_sequence A121208 A121209 
               A121210
%K A121207 nonn,tabl
%O A121207 0,6
%A A121207 N. J. A. Sloane (njas(AT)research.att.com), based on email from R. J. 
               Mathar, Dec 11 2006