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A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

%I A175840 #22 Jun 28 2015 18:16:23
%S A175840 1,3,2,9,6,4,27,18,12,8,81,54,36,24,16,243,162,108,72,48,32,729,486,
%T A175840 324,216,144,96,64,2187,1458,972,648,432,288,192,128,6561,4374,2916,
%U A175840 1944,1296,864,576,384,256,19683,13122,8748,5832,3888,2592,1728,1152,768,512
%N A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.
%C A175840 Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.
%C A175840 This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.
%H A175840 Reinhard Zumkeller, <a href="/A175840/b175840.txt">Rows n = 0..120 of triangle, flattened</a>
%H A175840 J. O'Connor and E.F. Robertson, <a href="http://www-history.mcs.st-and.ac.uk/history/Biographies/Nicomachus.html">Nicomachus of Gerasa</a>, The MacTutor History of Mathematics archive, 2010.
%H A175840 Jay Kappraff, <a href="http://www.nexusjournal.com">The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion</a>, Nexus Network Journal, vol. 2, no. 4 (October 2000).
%H A175840 Hendrik Lenstra, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2001-02-1-023.pdf">Aeternitatem Cogita</a>, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
%H A175840 Ivars Peterson, <a href="https://archive.is/iRXz">Medieval Harmony</a>, Math Trek, Mathematical Association of America, 1998.
%F A175840 T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.
%F A175840 T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.
%e A175840 1;
%e A175840 3, 2;
%e A175840 9, 6, 4;
%e A175840 27, 18, 12, 8;
%e A175840 81, 54, 36, 24, 16;
%e A175840 243, 162, 108, 72, 48, 32;
%p A175840 A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);
%t A175840 Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, May 08 2013 *)
%o A175840 (Haskell)
%o A175840 a175840 n k = a175840_tabf !! n !! k
%o A175840 a175840_row n = a175840_tabf !! n
%o A175840 a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
%o A175840 -- _Reinhard Zumkeller_, Jun 08 2013
%Y A175840 Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).
%Y A175840 Cf. A000079, A000244, A000400, A003586.
%K A175840 easy,nonn,tabl
%O A175840 0,2
%A A175840 _Johannes W. Meijer_, Sep 21 2010, Jul 13 2011, Jun 03 2012