%I
%S 1,3,3,6,9,6,10,18,17,10,15,30,33,27,15,21,45,54,51,39,21,28,63,80,82,
%T 72,53,28,36,84,111,120,114,96,69,36,45,108,147,165,165,150,123,87,45,
%U 55,135,188,217,225,215,190,153,107,55,66,165,234,276,294,291,270,234
%N The triangular triangle.
%C The nth row of the triangular table begins by considering n triangular numbers (A000217) in order. Now segregate them into n groups beginning with n members in the first group, n1 members in the second group, etc. Now sum each group. Thus the first term is the sum of first n numbers = n(n+1)/2, the second term is the sum of the next n1 terms (from n+1 to 2n1), the third term is the sum of the next n2 terms (2n to 3n3), etc. and the last term is simply n(n+1)/2. This triangle can be called a triangular triangle. The sequence contains the triangle by rows.
%H Reinhard Zumkeller, <a href="/A093445/b093445.txt">Rows n = 1..100 of triangle, flattened</a>
%F T(n) = A000217(n) is the nth Triangular number. TT(n, k) is the kth term of the nth row, 0 < k <= n.
%F TT(n, k) = T(k*n  T(k  1))  T((k  1)*n  T(k  2)).
%F TT(n, 1) = TT(n, n) = T(n) = A000217(n).
%e Triangle begins:
%e 1
%e 3, 3
%e 6, 9, 6
%e 10, 18, 17, 10
%e 15, 30, 33, 27, 15
%e 21, 45, 54, 51, 39, 21
%e 28, 63, 80, 82, 72, 53, 28
%e 36, 84, 111, 120, 114, 96, 69, 36
%e The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10.
%p A093445 := proc(n,k)
%p A000217(k*nA000217(k1))A000217((k1)*nA000217(k2)) ;
%p end proc:
%p seq(seq(A093445(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Dec 09 2015
%t T[n_] := n(n + 1)/2; TT[n_, k_] := T[k*n  T[k  1]]  T[(k  1)*n  T[k  2]]; Flatten[ Table[ TT[n, k], {n, 1, 11}, {k, 1, n}]] (* _Robert G. Wilson v_, Apr 24 2004 *)
%t Table[Total/@TakeList[Range[(n(n+1))/2],Range[n,1,1]],{n,20}]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Feb 15 2019 *)
%o (Haskell)
%o a093445 n k = a093445_row n !! (k1)
%o a093445_row n = f [n, n  1 .. 1] [1 ..] where
%o f [] _ = []
%o f (x:xs) ys = sum us : f xs vs where (us,vs) = splitAt x ys
%o a093445_tabl = map a093445_row [1 ..]
%o  _Reinhard Zumkeller_, Oct 03 2012
%Y Cf. A000217, A093446. TT(n, 2) = A045943. TT(n, n1) = A014209. TT(0, k) = A027480.
%Y Cf. A005920 (central terms), A002817 (row sums).
%K nonn,nice,tabl
%O 1,2
%A _Amarnath Murthy_, Apr 02 2004
%E Edited, corrected and extended by _Robert G. Wilson v_, Apr 24 2004
