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%I A003992 #50 Sep 08 2022 08:44:32
%S A003992 1,1,0,1,1,0,1,2,1,0,1,3,4,1,0,1,4,9,8,1,0,1,5,16,27,16,1,0,1,6,25,64,
%T A003992 81,32,1,0,1,7,36,125,256,243,64,1,0,1,8,49,216,625,1024,729,128,1,0,
%U A003992 1,9,64,343,1296,3125,4096,2187,256,1,0,1,10,81,512,2401,7776,15625,16384,6561,512,1,0
%N A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.
%C A003992 If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - _Robert A. Russell_, Nov 08 2018
%C A003992 T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - _Geoffrey Critzer_, Apr 03 2020
%H A003992 T. D. Noe, <a href="/A003992/b003992.txt">Rows n = 0..50 of triangle, flattened</a>
%F A003992 E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - _Paul D. Hanna_, Oct 22 2004
%F A003992 E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - _Franklin T. Adams-Watters_, Jun 23 2006
%e A003992 Rows begin:
%e A003992 [1, 0,  0,   0,    0,     0,      0,      0, ...],
%e A003992 [1, 1,  1,   1,    1,     1,      1,      1, ...],
%e A003992 [1, 2,  4,   8,   16,    32,     64,    128, ...],
%e A003992 [1, 3,  9,  27,   81,   243,    729,   2187, ...],
%e A003992 [1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
%e A003992 [1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
%e A003992 [1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
%e A003992 [1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...
%t A003992 Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten
%o A003992 (PARI) T(n,k) = (n-k)^k \\ _Charles R Greathouse IV_, Feb 07 2017
%o A003992 (Magma) [[(n-k)^k: k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 08 2018
%Y A003992 Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
%Y A003992 Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
%Y A003992 Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
%Y A003992 Cf. A099555.
%Y A003992 Transpose is A004248. See A051128, A095884, A009999 for other versions.
%Y A003992 Cf. A277504 (unoriented), A293500 (chiral).
%K A003992 easy,nice,nonn,tabl
%O A003992 0,8
%A A003992 _Marc LeBrun_
%E A003992 More terms from _David W. Wilson_
%E A003992 Edited by _Paul D. Hanna_, Oct 22 2004

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