A059300 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.

1, 1, 2, 1, 6, 3, 1, 12, 24, 4, 1, 20, 90, 80, 5, 1, 30, 240, 540, 240, 6, 1, 42, 525, 2240, 2835, 672, 7, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 1, 90, 2880, 41160, 272160, 787500, 860160, 262440, 11520, 10

Offset: 0

Author

N. J. A. Sloane, Jan 25 2001

Keywords

• nonn
• tabl

Examples

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

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
• John Riordan and N. J. A. Sloane, Correspondence, 1974

Crossrefs

There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.

Programs

Magma

/* As triangle: */ [[Binomial(n+1,n-k+1)*(n-k+1)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

Mathematica

t[n_, k_] := Binomial[n + 1, k]*(n - k + 1)^k; Flatten@Table[t[n, k], {n, 0, 9}, {k, 0, n}] (* Arkadiusz Wesolowski, Mar 23 2013 *)

PARI

for(n=0, 25, for(k=0, n, print1(binomial(n+1,k)*(n-k+1)^k, ", "))) \\ G. C. Greubel, Jan 05 2017

Formula

T(n,k) = binomial(n+1,n-k+1)*(n-k+1)^k. - R. J. Mathar, Mar 14 2013