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A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

%I A098546 #15 Jun 27 2025 19:14:22
%S A098546 1,2,1,3,3,1,4,6,6,4,1,5,10,10,10,10,5,1,6,15,15,15,20,20,20,15,15,6,
%T A098546 1,7,21,21,21,35,35,35,35,35,35,35,21,21,7,1,8,28,28,28,28,56,56,56,
%U A098546 56,56,70,70,70,70,70,56,56,56,28,28,8,1,9,36,36,36,36,84,84,84,84,84,84,84
%N A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.
%C A098546 A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end since A098346 times A049019 also yields A049009. (cf. A000312 and A000670).
%C A098546 Partitions are in Abramowitz and Stegun order. - _Franklin T. Adams-Watters_, Nov 20 2006
%H A098546 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A098546 a(n) = Combin( A036042(n), A036043(n) )
%e A098546 A036042 begins 1 2 2 3 3 3 4 4 4 4 4 ...
%e A098546 A036043 begins 1 1 2 1 2 3 1 2 2 3 4 ...
%e A098546 so a(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
%e A098546 Table begins
%e A098546 .
%e A098546 1
%e A098546 2 1
%e A098546 3 3  1
%e A098546 4 6  6  4  1
%e A098546 5 10 10 10 10 5  1
%e A098546 6 15 15 20 15 20 15 20 15 6 1
%e A098546 .
%t A098546 Table[Sequence @@
%t A098546   Map[Function[p, Binomial[n, Length[p]]], IntegerPartitions[n]], {n,
%t A098546   1, 10}] (* _Olivier Gérard_, May 07 2024 *)
%Y A098546 Cf. A090657, A000041 (row lengths), A098545 (row sums), A036036, A036042, A036043.
%K A098546 easy,nonn,tabf
%O A098546 1,2
%A A098546 _Alford Arnold_, Sep 14 2004