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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
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
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 July 12 15:21 EDT 2024. Contains 374251 sequences. (Running on oeis4.)