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

2,2

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.

Links

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

Formula

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

Example

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

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 *)

Prog

(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
(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
(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
(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

Crossrefs

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

Sequence in context: A215942 A054908 A141826 * A271696 A271090 A271284

Adjacent sequences: A178753 A178754 A178755 * A178757 A178758 A178759

Keyword

nonn,tabl

Author

Geoffrey Critzer, Dec 26 2010

Status

approved