A156354 Triangle T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1, read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1

Offset: 0

Author

Roger L. Bagula, Feb 08 2009

Keywords

• nonn
• tabl

Comments

This sequence is an approximation of Pascal's triangle with interior Kurtosis.

Essentially the same as A055652. - R. J. Mathar, Feb 19 2009

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,   8,    4,     1;
  1,  5,  17,   17,     5,     1;
  1,  6,  32,   54,    32,     6,     1;
  1,  7,  57,  145,   145,    57,     7,     1;
  1,  8, 100,  368,   512,   368,   100,     8,    1;
  1,  9, 177,  945,  1649,  1649,   945,   177,    9,   1;
  1, 10, 320, 2530,  5392,  6250,  5392,  2530,  320,  10,  1;
  1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1;
The interior Kurtosis, T(n,k) - binomial(n, k), is:
  0;
  0, 0;
  0, 0,   0;
  0, 0,   0,    0;
  0, 0,   2,    0,     0;
  0, 0,   7,    7,     0,     0;
  0, 0,  17,   34,    17,     0,     0;
  0, 0,  36,  110,   110,    36,     0,     0;
  0, 0,  72,  312,   442,   312,    72,     0,    0;
  0, 0, 141,  861,  1523,  1523,   861,   141,    0,   0;
  0, 0, 275, 2410,  5182,  5998,  5182,  2410,  275,   0, 0;
  0, 0, 538, 6908, 18455, 22939, 22939, 18455, 6908, 538, 0, 0;
		

Links

• G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Crossrefs

Cf. A026898.

Programs

Magma

[k eq 0 select 1 else k^(n-k) + (n-k)^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2021

Mathematica

T[n_, k_]:= If[n==0, 1, (k^(n-k) + (n-k)^k)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

Sage

flatten([[1 if k==n else k^(n-k) + (n-k)^k for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 07 2021

Formula

T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1.

T(n, k) = T(n, n-k).

Sum_{k=0..n} T(n,k) = [n=0] + 2*A026898(n-1). - G. C. Greubel, Mar 07 2021

Extensions

Edited by G. C. Greubel, Mar 07 2021