0,4

Or, prefix Aitken's array (A011971) with a leading diagonal of zeros and take the differences of each row to get the new triangle.

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

Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15...

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

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

Triangle starts:

1;

1,1;

2,1,2;

5,2,3,5;

15,5,7,10,15;

52,15,20,27,37,52;

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

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

Cf. A000110, A005493, A011971.

Sequence in context: A153206 A144155 A109631 this_sequence A064192 A124218 A025165

Adjacent sequences: A095146 A095147 A095148 this_sequence A095150 A095151 A095152

nonn,tabl,easy,nice

Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2004

Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2007

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jun 01 2005, Jun 16 2007