A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 24, 4, 0, 1, 20, 90, 80, 5, 0, 1, 30, 240, 540, 240, 6, 0, 1, 42, 525, 2240, 2835, 672, 7, 0, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 0, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 0, 1, 90, 2880, 41160

Offset

0,5

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Example

Triangle begins:
1,
1,  0,
1,  2,   0,
1,  6,   3,    0,
1, 12,  24,    4,    0,
1, 20,  90,   80,    5,   0,
1, 30, 240,  540,  240,   6, 0,
1, 42, 525, 2240, 2835, 672, 7, 0,
...

Maple

T := (n, k) -> binomial(n, k) * (n - k)^k:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

Mathematica

t[n_, k_] := Binomial[n, k]*(n - k)^k; Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

Prog

(Magma) /* As triangle: */ [[Binomial(n, k)*(n-k)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015
(PARI) concat([1], for(n=0, 25, for(k=0, n, print1(binomial(n, k)*(n-k)^k, ", ")))) \\ G. C. Greubel, Jan 05 2017

Crossrefs

There are 4 versions: A059297-A059300.

Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.

Row sums are A000248.

Sequence in context: A339031 A367270 A365770 * A332673 A128722 A324659

Adjacent sequences: A059296 A059297 A059298 * A059300 A059301 A059302

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 25 2001

Extensions

Name corrected by Peter Luschny, Nov 12 2023

Status

approved