cp's OEIS Frontend

A298804 Triangle T(n,k) (1 <= k <= n) read by rows: A046936 with rows reversed and offset changed to 1.

%I A298804 #14 Jul 09 2025 04:46:38
%S A298804 0,1,1,3,2,1,9,6,4,3,31,22,16,12,9,121,90,68,52,40,31,523,402,312,244,
%T A298804 192,152,121
%N A298804 Triangle T(n,k) (1 <= k <= n) read by rows: A046936 with rows reversed and offset changed to 1.
%C A298804 This is another version of Moser's version (A046936) of Aitken's array (A011971).
%C A298804 Although offset 0 is better for A011971 and A046936, for this version offset 1 is more appropriate.
%C A298804 Comments from _Don Knuth_, Jan 29 2018 (Start):
%C A298804 a(n,k) is the number of set partitions (i.e. equivalence classes) in which (i) 1 is not equivalent to 2, ..., nor k; and (ii) the last part, when parts are ordered by their smallest element, has size 1; (iii) that last part isn't simply "1". (Equivalently, n>1.)
%C A298804 It's not difficult to prove this characterization of a(k,n). For example, if we know that there are 22 partitions of {1,2,3,4,5} with 1 inequivalent to 2, and 6 partitions of {1,2,3,4} with
%C A298804 1 inequivalent to 2, then there are 6 partitions of {1,2,3,4,5} with 1 inequivalent to 2 and 1 equivalent to 3. Hence there are 16 with 1 equivalent to neither 2 nor 3.
%C A298804 The same property, but leaving out conditions (ii) and (iii), characterizes Pierce's triangular array A123346.  (End)
%H A298804 Don Knuth, <a href="/A040027/a040027.txt">Email to N. J. A. Sloane</a>, Jan 29 2018
%e A298804 Triangle begins:
%e A298804 0,
%e A298804 1, 1,
%e A298804 3, 2, 1,
%e A298804 9, 6, 4, 3,
%e A298804 31, 22, 16, 12, 9,
%e A298804 121, 90, 68, 52, 40, 31
%e A298804 523, 402, 312, 244, 192, 152, 121
%e A298804 ...
%Y A298804 Cf. A011971, A040027, A046936, A123346.
%K A298804 nonn,tabl,more
%O A298804 1,4
%A A298804 _N. J. A. Sloane_, Jan 30 2018, following a suggestion from _Don Knuth_, Jan 29 2018