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A178756 Rectangular array T(n,k) = binomial(n,2)*k*n^(k-1) read by antidiagonals. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 11 18:01 EDT 2020. Contains 335652 sequences. (Running on oeis4.)