OFFSET
1,2
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).
KEYWORD
nonn,tabl
AUTHOR
Ross La Haye, Jan 22 2008
STATUS
approved