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
Examples
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
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).
Programs
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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 *)
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