%I A035206
%S A035206 1,2,1,3,6,1,4,12,6,12,1,5,20,20,30,30,20,1,6,30,30,15,60,120,20,60,90,
%T A035206 30,1,7,42,42,42,105,210,105,105,140,420,140,105,210,42,1,8,56,56,56,28,
%U A035206 168,336,336,168,168,280,840,420,840,70,280,1120,560,168,420,56,1,9,72
%N A035206 Number of multisets associated with least integer of each prime signature.
%C A035206 Multiplying by 1; 1,2; 1,3,6; 1,4,6,12,24; ... (A036038) yields 1; 2,
               2; 3,18,6; 4,48,36,144,24; ... in which the groups sum to 1; 4; 27; 
               256; .... (A000312).
%C A035206 a(n,k) enumerates distributions of n identical objects (balls) into m 
               of alltogether n distinguishable boxes. The k-th partition of n, 
               taken in the Abramowitz-Stegun (A-St) order, specifies the occupation 
               of the m =m(n,k)= A036043(n,k) boxes. m = m(n,k) is the number of 
               parts of the k-th partition of n. For the A-St ordering see pp.831-2 
               of the reference given in A117506. W. Lang, Nov 13 2007.
%C A035206 The sequence of row lengths is p(n)= A000041(n) (partition numbers) [1, 
               2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A035206 For the A-St order of partitions see the Abramowitz-Stegun reference 
               given in A117506.
%H A035206 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A035206 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A035206.text">
               First 10 rows and more. </a>
%F A035206 a(n,k) = A048996(n,k)*binomial(n,m(n,k)),n>=1, k=1,...,p(n) and m(n,k):=A036043(n,
               k) gives the number of parts of the k-th partition of n.
%e A035206 1; 2,1; 3,6,1; 4,12,6,12,1; 5,20,20,30,30,20,1; ...
%e A035206 a(5,5) relates to the partition (1,2^2) of n=5. Here m=3 and 5 indistinguishable 
               (identical)
%e A035206 balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with 
               one ball and two with two balls.
%e A035206 Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. W. Lang, Nov 
               13 2007.
%Y A035206 Cf. A036038, A048996, A049009.
%Y A035206 Cf. A001700 (row sums).
%Y A035206 Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the 
               numbers with like part numbers m).
%Y A035206 Sequence in context: A078760 A103280 A046899 this_sequence A115196 A093346 
               A115597
%Y A035206 Adjacent sequences: A035203 A035204 A035205 this_sequence A035207 A035208 
               A035209
%K A035206 nonn,tabf,easy
%O A035206 1,2
%A A035206 Alford Arnold (Alford1940(AT)aol.com)
%E A035206 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), 
               Jul 27 2006