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

0,5

Comments

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

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

Links

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

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

Example

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;

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

Prog

(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
(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

Crossrefs

Cf. A026898.

Sequence in context: A125806 A347148 A202756 * A295205 A297020 A099597

Adjacent sequences: A156351 A156352 A156353 * A156355 A156356 A156357

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 08 2009

Extensions

Edited by G. C. Greubel, Mar 07 2021

Status

approved