A178756 Rectangular array T(n,k) = binomial(n,2)*k*n^(k-1) read by antidiagonals.

1, 4, 3, 12, 18, 6, 32, 81, 48, 10, 80, 324, 288, 100, 15, 192, 1215, 1536, 750, 180, 21, 448, 4374, 7680, 5000, 1620, 294, 28, 1024, 15309, 36864, 31250, 12960, 3087, 448, 36, 2304, 52488, 172032, 187500, 97200, 28812, 5376, 648, 45

Offset: 2

Author

Geoffrey Critzer, Dec 26 2010

Keywords

• nonn
• tabl

Comments

T(n,k) is the sum of the digits in all n-ary words of length k. That is, sequences of k digits taken on an alphabet of {0,1,2,...,n-1}.

Note the rectangle is indexed begining from n = 2 (binary sequences) which is A001787.

Examples

			1,4,12,32,80,192,448,1024
3,18,81,324,1215,4374,15309,52488
6,48,288,1536,7680,36864,172032,786432
10,100,750,5000,31250,187500,1093750,6250000
15,180,1620,12960,97200,699840,4898880,33592320
		

Links

• G. C. Greubel, Antidiagonals n=2..101, flattened

Crossrefs

Cf. A036290 (ternary sequences), A034967 (decimal digits).

Programs

GAP

T:=Flat(List([3..10], n-> List([2..n-1], k-> Binomial(k,2)*(n-k)* k^(n-k-1) ))); # G. C. Greubel, Jan 24 2019

Magma

[[Binomial(k,2)*(n-k)*k^(n-k-1): k in [2..n-1]]: n in [3..10]]; // G. C. Greubel, Jan 24 2019

Maple

T:= (n, k)-> binomial(n, 2)*k*n^(k-1):
seq(seq(T(n, 1+d-n), n=2..d), d=2..14); # Alois P. Heinz, Jan 17 2013

Mathematica

Table[Range[8]! Rest[CoefficientList[Series[Binomial[n,2]x Exp[n x],{x,0,8}],x]],{n,2,10}]//Grid
T[n_, k_]:= Binomial[n, 2]*k*n^(k-1); Table[T[k,n-k], {n,2,10}, {k, 2, n-1}]//Flatten (* G. C. Greubel, Jan 24 2019 *)

PARI

{T(n,k) = binomial(n,2)*k*n^(k-1)};
for(n=2,10, for(k=2,n-1, print1(T(k,n-k), ", "))) \\ G. C. Greubel, Jan 24 2019

Sage

[[binomial(k,2)*(n-k)*k^(n-k-1) for k in (2..n-1)] for n in (3..10)] # G. C. Greubel, Jan 24 2019

Formula

E.g.f. for row n: binomial(n,2)*x*exp(n*x).