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A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals. 0
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
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
KEYWORD
nonn,tabl
AUTHOR
Ross La Haye, Jan 22 2008
STATUS
approved

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Last modified July 21 22:27 EDT 2024. Contains 374478 sequences. (Running on oeis4.)