A125234 -id:A125234 - cp's OEIS frontend

A126277 Triangle generated from Eulerian numbers.

1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 11, 15, 5, 1, 6, 15, 26, 31, 6, 1, 7, 19, 37, 57, 63, 7, 1, 8, 23, 48, 83, 120, 127, 8, 1, 9, 27, 59, 109, 177, 247, 255, 9, 1, 10, 31, 70, 135, 234, 367, 502, 511, 10

Offset: 1

Gary w. Adamson, Dec 23 2006

N-th diagonal starting from the right = binomial transform of [1, N, q, q, q, ...) where q = 2*N - 2. Given the infinite set of triangles "T" composed of partial column sums of the polygonal numbers, the N-th diagonal starting from the right = row sums of triangle "T": (T=3 = A104712; T=4 = A125165; T=5 = A125232; T=6 = A125233; T=7 = A125234, T=8 = A125235; and so on). For example, 3rd diagonal from the right = the offset Eulerian numbers, (1, 4, 11, 26, 57, 120, ...) = row sums of Triangle A104712 having partial column sums of the triangular numbers: 1; 3, 1; 6, 4, 1; 10, 10, 5, 1; 15, 20, 15, 6, 1; ... Row sums = A124671: (1, 3, 7, 16, 37, 85, 191, ...).

			First few rows of the triangle:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  7,  4;
  1,  5, 11, 15,   5;
  1,  6, 15, 26,  31,   6;
  1,  7, 19, 37,  57,  63,   7;
  1,  8, 23, 48,  83, 120, 127,   8;
  1,  9, 27, 59, 109, 177, 247, 255,   9;
  1, 10, 31, 70, 135, 234, 367, 502, 511, 10;
  ...
T(7,4) = 37 = A000295(4) + T(6,4) = 11 + 26.

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A126268, A000295, A104712, A125165, A104712, A125232 - A125235.

T[n_,1]:=1; T[n_,n_]:=n; T[n_,k_]:= T[n-1,k] + 2^k - k - 1; Table[T[n,k], {n,1,15}, {k,1,n}]//Flatten (* G. C. Greubel, Oct 23 2018 *)

{T(n,k) = if(k==1, 1, if(k==n, n, 2^k - k - 1 + T(n-1,k)))};
for(n=1,10, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 23 2018

Given right border = (1,2,3,...), T(n,k) = A000295(k) + T(n-1,k); where A000295 = the Eulerian numbers starting (0, 1, 4, 11, 26, 57, ...).

A125235 Triangle with the partial column sums of the octagonal numbers.

Original entry on oeis.org

1, 8, 1, 21, 9, 1, 40, 30, 10, 1, 65, 70, 40, 11, 1, 96, 135, 110, 51, 12, 1, 133, 231, 245, 161, 63, 13, 1, 176, 364, 476, 406, 224, 76, 14, 1, 225, 540, 840, 882, 630, 300, 90, 15, 1, 280, 765, 1380, 1722, 1512, 930, 390, 105, 16, 1

Offset: 1

Gary W. Adamson, Nov 24 2006

"Partial column sums" means the octagonal numbers are the 1st column, the 2nd column are the partial sums of the 1st column, the 3rd column are the partial sums of the 2nd, etc.

Row sums are 1, 9, 31, 81, 187, 405, 847 = 7*(2^n-1) - 6*n. - R. J. Mathar, Sep 06 2011

			First few rows of the triangle:
   1;
   8,   1;
  21,   9,   1;
  40,  30,  10,   1;
  65,  70,  40,  11,   1;
  96, 135, 110,  51,  12,   1;
  ...

Albert H. Beiler, Recreations in the Theory of Numbers, Dover (1966), p. 189.

Cf. A000567, A002414, A002419, A051843, A027810, A125232 - A125234.

t(n, k) = if (n <0, 0, if (k==1, n*(3*n-2), if (k > 1, t(n-1,k-1) + t(n-1,k))));
tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(t(n, k), ", ");); print(););} \\ Michel Marcus, Mar 04 2014

T(n,1) = A000567(n).
T(n,k) = T(n-1,k-1) + T(n-1,k), k>1.
T(n,2) = A002414(n-1).
T(n,3) = A002419(n-2).
T(n,4) = A051843(n-4).
T(n,5) = A027810(n-6).

More terms from Michel Marcus, Mar 04 2014

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A126277 Triangle generated from Eulerian numbers.

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