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%I #27 Apr 28 2016 11:29:32
%S 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 90,30,1,7,42,42,42,105,210,105,245,420,140,105,210,42,1,8,56,56,224,
%U 28,336,336,280,168,168,840,420,1120,70,1120,560,168,420,56,1
%N Triangle of multiplicities D(n) of multinomial coefficients corresponding to sequence A210237.
%C Multiplicity D(n) of multinomial coefficient M(n) is the number of ways the same value of M(n)=n!/(m1!*m2!*..*mk!) is obtained by distributing n identical balls into k distinguishable bins.
%C Differs from A209936 after a(21).
%C Differs from A035206 after a(36).
%C The checksum relationship: sum(M(n)*D(n)) = k^n
%C The number of terms per row (for each value of n starting with n=1) forms sequence A070289.
%H Sergei Viznyuk, <a href="http://phystech.com/ftp/s_A210238.c">C-program</a> for the sequence.
%e 1
%e 2, 1
%e 3, 6, 1
%e 4, 12, 6, 12, 1
%e 5, 20, 20, 30, 30, 20, 1
%e 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1
%e 7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1
%e Thus for n=3 (third row) the same value of multinomial coefficient follows from the following combinations:
%e 3!/(3!0!0!) 3!/(0!3!0!) 3!/(0!0!3!) (i.e. multiplicity=3)
%e 3!/(2!1!0!) 3!/(2!0!1!) 3!/(0!2!1!) 3!/(0!1!2!) 3!/(1!0!2!) 3!/(1!2!0!) (i.e. multiplicity=6)
%e 3!/(1!1!1!) (i.e. multiplicity=1)
%t Table[Last/@Tally[Multinomial@@@Compositions[k,k]],{k,8}] (* _Wouter Meeussen_, Mar 09 2013 *)
%Y Cf. A210237, A209936, A035206, A070289.
%K nonn,tabf
%O 1,2
%A _Sergei Viznyuk_, Mar 18 2012
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