A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals.

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

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