A093446 -id:A093446 - cp's OEIS frontend

A093445 The triangular triangle.

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, 72, 53, 28, 36, 84, 111, 120, 114, 96, 69, 36, 45, 108, 147, 165, 165, 150, 123, 87, 45, 55, 135, 188, 217, 225, 215, 190, 153, 107, 55, 66, 165, 234, 276, 294, 291, 270, 234

Offset: 1

Amarnath Murthy, Apr 02 2004

The n-th 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, n-1 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 n-1 terms (from n+1 to 2n-1), the third term is the sum of the next n-2 terms (2n to 3n-3), 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.

			Triangle begins:
   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,  72, 53, 28
  36, 84, 111, 120, 114, 96, 69, 36
The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A000217, A093446. TT(n, 2) = A045943. TT(n, n-1) = A014209. TT(0, k) = A027480.

Cf. A005920 (central terms), A002817 (row sums).

a093445 n k = a093445_row n !! (k-1)
a093445_row n = f [n, n - 1 .. 1] [1 ..] where
   f [] _      = []
   f (x:xs) ys = sum us : f xs vs where (us,vs) = splitAt x ys
a093445_tabl = map a093445_row [1 ..]
-- Reinhard Zumkeller, Oct 03 2012

A093445 := proc(n,k)
    A000217(k*n-A000217(k-1))-A000217((k-1)*n-A000217(k-2)) ;
end proc:
seq(seq(A093445(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Dec 09 2015

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 *)
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 *)

T(n) = A000217(n) is the n-th Triangular number. TT(n, k) is the k-th term of the n-th row, 0 < k <= n.
TT(n, k) = T(k*n - T(k - 1)) - T((k - 1)*n - T(k - 2)).
TT(n, 1) = TT(n, n) = T(n) = A000217(n).

Edited, corrected and extended by Robert G. Wilson v, Apr 24 2004

A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)n-T(k-2)+1 through kn-T(k-1).

Original entry on oeis.org

1, 2, 3, 6, 20, 6, 24, 210, 72, 10, 120, 3024, 1320, 182, 15, 720, 55440, 32760, 4896, 380, 21, 5040, 1235520, 1028160, 175560, 13800, 702, 28, 40320, 32432400, 39070080, 7893600, 657720, 32736, 1190, 36, 362880, 980179200, 1744364160

Offset: 1

Amarnath Murthy, Apr 02 2004

This is built by starting from the sequence 1,2,....,T(n) in row n, where T(n) is the triangular number A000217(n) and packaging its first n, the next n-1, the next n-2,... up to the last number in groups and writing down the product of each group in one cell of the triangle. The first column is A000142. The second column is essentially A006963. The 3rd column is essentially A001763. The diagonal is A000217. - R. J. Mathar, Jul 26 2007

			In factorized notation the triangle starts
1;
1*2, 3;
1*2*3, 4*5, 6;
1*2*3*4, 5*6*7, 8*9, 10;
1*2*3*4*5, 6*7*8*9, 10*11*12, 13*14, 15;
which gives
1;
2, 3;
6, 20, 6;
24, 210, 72, 10;
120, 3024, 1320, 182, 15;
720,55440,32760, 4896, 380, 21;

Cf. A093445, A093446, A093448.

A000217 := proc(n) n*(n+1)/2 ; end: A093447 := proc(n,k) factorial(k*n-A000217(k-1))/factorial((k-1)*n-A000217(k-2)) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d, ",A093447(n,k)) ; od ; od: # R. J. Mathar, Jul 26 2007

a(n,k)= [k*n-T(k-1)]!/[(k-1)*n-T(k-2)]! where T(n)=A000217(n). - R. J. Mathar, Jul 26 2007

More terms from R. J. Mathar, Jul 26 2007

A093448 Rows sums of the triangle A093447.

Original entry on oeis.org

1, 5, 32, 316, 4661, 94217, 2458810, 80128082, 3193424921, 153067911301, 8685693546692, 574691476630760, 43735137898763917, 3784250198022172001, 368841694500041857646, 40194470526005627873182

Offset: 1

Amarnath Murthy, Apr 02 2004

nonn

			The row for n = 4 is
(1*2*3*4), (5*6*7), (8*9), 10 or
24 210 72 10.
hence a(4) = 24 +210 +72 +10 = 316.

Cf. A093445, A093446, A093447.

A000217 := proc(n) n*(n+1)/2 ; end: A093447 := proc(n,k) factorial(k*n-A000217(k-1))/factorial((k-1)*n-A000217(k-2)) ; end: A093448 := proc(n) add( A093447(n,k),k=1..n) ; end: for n from 1 to 26 do printf("%d, ",A093448(n)) ; od: # R. J. Mathar, Jul 27 2007

More terms from R. J. Mathar, Jul 27 2007

cp's OEIS Frontend

A093445 The triangular triangle.

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A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)n-T(k-2)+1 through kn-T(k-1).

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A093448 Rows sums of the triangle A093447.

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cp's OEIS Frontend

A093445 The triangular triangle.

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A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)*n-T(k-2)+1 through k*n-T(k-1).

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A093448 Rows sums of the triangle A093447.

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Author

Keywords

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Maple

Extensions

A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)n-T(k-2)+1 through kn-T(k-1).