%I A095149
%S A095149 1,1,1,2,1,2,5,2,3,5,15,5,7,10,15,52,15,20,27,37,52,203,52,67,87,114,
%T A095149 151,203,877,203,255,322,409,523,674,877,4140,877,1080,1335,1657,2066,
%U A095149 2589,3263,4140,21147,4140,5017,6097,7432,9089,11155,13744,17007,21147
%N A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal 
               before it given by the Bell numbers (A000110), 1,1,2,5,15,52,...
%C A095149 Or, prefix Aitken's array (A011971) with a leading diagonal of zeros 
               and take the differences of each row to get the new triangle.
%C A095149 With offset 1, triangle read by rows: T(n,k) is the number of partitions 
               of the set {1,2,...,n} in which k is the largest entry in the block 
               containing 1 (1<=k<=n). For example - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Oct 29 2006
%C A095149 Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15...
%F A095149 With offset 1, T(n,1)=T(n,n)=T(n+1,2)=B(n-1)=A000110(n-1) (the Bell numbers). 
               T(n,k)=T(n,k-1)+T(n-1,k-1) for n>=k>=3. T(n,n-1)=B(n-1)-B(n-2)=A005493(n-3) 
               for n>=3 (B(q) are the Bell numbers A000110). T(n,k)=A011971(n-2,
               k-2) for n>=k>=2. In other words, deleting first row and first column 
               we obtain Aitken's array A011971 (also called Bell or Pierce triangle; 
               offset in A011971 is 0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Oct 29 2006
%F A095149 T(n,1)=B(n-1); T(n,2)=B(n-2) for n>=2; T(n,k)=Sum(binom(k-2,i)*B(n-2-i), 
               i=0..k-2) for 3<=k<=n, where B(q) are the Bell numbers (A000110). 
               Generating polynomial of row n is P[n](t)=Q[n](t,1), where Q[n](t,
               s)=t^n*Q[n-1](1,s)+s*dQ[n-1](t,s)/ds +(s-1) Q[n-1](t,s) ; Q[1](t,
               s)=ts. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2006
%e A095149 Triangle starts:
%e A095149 1;
%e A095149 1,1;
%e A095149 2,1,2;
%e A095149 5,2,3,5;
%e A095149 15,5,7,10,15;
%e A095149 52,15,20,27,37,52;
%p A095149 A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else 
               A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi 
               ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else 
               add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 
               := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,
               k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, 
               ",A095149(n,k)) ; od ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Feb 05 2007
%p A095149 with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 
               then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) 
               else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) 
               od; # yields sequence in triangular form Q[1]:=t*s: for n from 2 
               to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) 
               od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 
               1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular 
               form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2006
%Y A095149 Cf. A000110, A005493, A011971.
%Y A095149 Sequence in context: A153206 A144155 A109631 this_sequence A064192 A124218 
               A025165
%Y A095149 Adjacent sequences: A095146 A095147 A095148 this_sequence A095150 A095151 
               A095152
%K A095149 nonn,tabl,easy,nice
%O A095149 0,4
%A A095149 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2004
%E A095149 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Feb 05 2007
%E A095149 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jun 01 2005, 
               Jun 16 2007