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A156354 Triangle T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1, read by rows. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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: A114202 A125806 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

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Last modified August 5 21:45 EDT 2021. Contains 346488 sequences. (Running on oeis4.)