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%I #29 Aug 08 2022 11:56:39
%S 1,1,2,1,3,6,1,4,12,6,12,1,5,20,20,30,30,20,1,6,30,30,15,60,120,20,60,
%T 90,30,1,7,42,42,42,105,210,105,105,140,420,140,105,210,42,1,8,56,56,
%U 56,28,168,336,336,168,168,280,840,420,840,70,280,1120,560,168,420,56,1,9,72
%N Number of multisets associated with least integer of each prime signature.
%C a(n,k) multiplied by A036038(n,k) yields A049009(n,k).
%C a(n,k) enumerates distributions of n identical objects (balls) into m of altogether 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. - _Wolfdieter Lang_, Nov 13 2007.
%C The sequence of row lengths is p(n)= A000041(n) (partition numbers).
%C For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%C The corresponding triangle with summed row entries which belong to partitions of the same number of parts k is A103371. [_Wolfdieter Lang_, Jul 11 2012]
%H Andrew Howroyd, <a href="/A035206/b035206.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H 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].
%H Wolfdieter Lang, <a href="/A035206/a035206.pdf">First 10 rows and more.</a>
%F 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 n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e 0 1
%e 1 1
%e 2 2 1
%e 3 3 6 1
%e 4 4 12 6 12 1
%e 5 5 20 20 30 30 20 1
%e 6 6 30 30 15 60 120 20 60 90 30 1
%e 7 7 42 42 42 105 210 105 105 140 420 140 105 210 42 1
%e ...
%e Row No. 8: 8 56 56 56 28 168 336 336 168 168 280 840 420 840 70 280 1120 560 168 420 56 1
%e Row No. 9: 9 72 72 72 72 252 504 504 252 252 504 84 504 1512 1512 1512 1512 504 630 2520 1260 3780 630 504 2520 1680 252 756 72 1
%e [rewritten and extended table by _Wolfdieter Lang_, Jul 11 2012]
%e a(5,5) relates to the partition (1,2^2) of n=5. Here m=3 and 5 indistinguishable (identical) balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with one ball and two with two balls.
%e Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. _Wolfdieter Lang_, Nov 13 2007.
%o (PARI)
%o C(sig)={my(S=Set(sig)); binomial(vecsum(sig), #sig)*(#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
%o Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
%o { for(n=0, 7, print(Row(n))) } \\ _Andrew Howroyd_, Oct 18 2020
%Y Cf. A036038, A048996, A049009.
%Y Cf. A001700 (row sums).
%Y Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).
%K nonn,tabf,easy
%O 0,3
%A _Alford Arnold_
%E More terms from _Joshua Zucker_, Jul 27 2006
%E a(0)=1 prepended by _Andrew Howroyd_, Oct 18 2020
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