# Peirce Numbers

N. J. A. Sloane njas at research.att.com
Tue Mar 12 03:21:58 CET 2002

```Jon said:
no, sorry, the array here looks like:

|   1
|   2   1
|   5   3   2
|  15  10   7   5
|  52  37  27  20  15
| 203 151 114  87  67  52

but i seem to remember this somewhere?

Me:  This is Aitken's array, see A011971:

%I A011971
%S A011971 1,1,2,2,3,5,5,7,10,15,15,20,27,37,52,52,67,87,114,151,203,203,255,
%T A011971 322,409,523,674,877,877,1080,1335,1657,2066,2589,3263,4140,4140,
%U A011971 5017,6097,7432,9089,11155,13744,17007,21147,21147,25287,30304
%N A011971 Aitken's array: triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=a(n-1,n-1), a(n,k)=a(n,k-1)+a(n-1,k-1).
%D A011971 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 212.
%D A011971 Leo Moser showed this to Richard Guy in March 1968.
%e A011971 1; 1,2; 2,3,5; 5,7,10,15; 15,20,27,37,52; ...
%p A011971 A011971:=proc(n,k) option remember; if n=0 and k=0 then 1 elif k=0 then A011971(n-1,n-1) else A011971(n,k-1)+A011971(n-1,k-1); fi: end;
%p A011971 for n from 0 to 12 do lprint([ seq(A011971(n,k),k=0..n) ]); od:
%Y A011971 Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc.; A011968, A011969. Cf. A046934, A011972.
%K A011971 tabl,nonn,easy,nice
%O A011971 0,3
%A A011971 njas, J. H. Conway, Richard Guy

NJAS

```