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A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals. 2
1, 2, 2, 3, 8, 3, 4, 24, 18, 4, 5, 64, 81, 32, 5, 6, 160, 324, 192, 50, 6, 7, 384, 1215, 1024, 375, 72, 7, 8, 896, 4374, 5120, 2500, 648, 98, 8, 9, 2048, 15309, 24576, 15625, 5184, 1029, 128, 9, 10, 4608, 52488, 114688, 93750, 38880, 9604, 1536, 162, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..55.

FORMULA

a(n,k) = n*k^n.

O.g.f. (by columns): (k*x)/(-1+k*x)^2.

E.g.f. (by columns): k*x*exp(k*x).

a(n,k) = Sum[k^n,{j,1,n}] = n*Sum[C(n,m)*(k-1)^m,{m,0,n}]. - Ross La Haye, Jan 26 2008

EXAMPLE

a(2,2) = 8 because there are 2^2 = 4 2-length words on a 2 letter alphabet, each of size 2 and 2*4 = 8.

Array begins:

==================================================================

n\k|  1     2       3        4         5         6          7  ...

---|--------------------------------------------------------------

1  |  1     2       3        4         5         6          7  ...

2  |  2     8      18       32        50        72         98  ...

3  |  3    24      81      192       375       648       1029  ...

4  |  4    64     324     1024      2500      5184       9604  ...

5  |  5   160    1215     5120     15625     38880      84035  ...

6  |  6   384    4374    24576     93750    279936     705894  ...

7  |  7   896   15309   114688    546875   1959552    5764801  ...

8  |  8  2048   52488   524288   3125000  13436928   46118408  ...

9  |  9  4608  177147  2359296  17578125  90699264  363182463  ...

... - Franck Maminirina Ramaharo, Aug 07 2018

MATHEMATICA

t[n_, k_] := Sum[k^n, {j, n}]; Table[ t[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 07 2018 *)

CROSSREFS

Cf. a(n, 1) = a(1, k) = A000027(n); a(n, 2) = A036289(n); a(n, 3) = A036290(n); a(n, 4) = A018215(n); a(n, 5) = A036291(n); a(n, 6) = A036292(n); a(n, 7) = A036293(n); a(n, 8) = A036294(n); a(2, k) = A001105(k); a(3, k) = A117642(k); a(n, n) = A007778(n); a(n, n+1) = A066274(n+1): sum[a(i-1, n-i+1), {i, 1, n}] = A062807(n).

Sequence in context: A186753 A135835 A177696 * A141617 A267644 A204197

Adjacent sequences:  A134571 A134572 A134573 * A134575 A134576 A134577

KEYWORD

nonn,tabl

AUTHOR

Ross La Haye, Jan 22 2008

STATUS

approved

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Last modified May 9 22:17 EDT 2021. Contains 343746 sequences. (Running on oeis4.)