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A055134 Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.

%I A055134 #48 Feb 16 2025 08:32:42
%S A055134 1,0,1,1,2,1,8,12,6,1,81,108,54,12,1,1024,1280,640,160,20,1,15625,
%T A055134 18750,9375,2500,375,30,1,279936,326592,163296,45360,7560,756,42,1,
%U A055134 5764801,6588344,3294172,941192,168070,19208,1372,56,1,134217728
%N A055134 Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.
%C A055134 The same triangle (except for signs) may be obtained from the determinants of the Brahmagupta matrices, setting x->Sqrt[z], y->1, t->n. - _Roger L. Bagula_, Apr 09 2008
%C A055134 From _Bob Selcoe_, Nov 15 2014 (Start):
%C A055134 T(n,k)/A000312(n) is the probability P(n,k) that any member (j) of set J={1..n} will be selected k times given n random draws from J. This is equivalent to rolling an n-sided die (with standard assumptions) with sides numbered j=1..n: P(n,k) is the probability that any j will show k times with n rolls.
%C A055134 P(n,k) = (n-2)!*(n-1)^(n-k+1 )/k!*(n-k)!*n^(n-1); n>1. As n approaches infinity, P(n,0) and P(n,1) approach 1/e. (End)
%C A055134 Row sums give n^n (see A000312). - _Bob Selcoe_, Sep 08 2015
%H A055134 G. C. Greubel, <a href="/A055134/b055134.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A055134 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BrahmaguptaMatrix.html">Brahmagupta Matrix</a>.
%F A055134 T(n, k) = C(n, k)*(n-1)^(n-k), for n>1.
%F A055134 E.g.f.: (-LambertW(-y)/y)^(x-1)/(1+LambertW(-y)). - _Vladeta Jovovic_
%F A055134 O.g.f. for row n: (x + n - 1)^n. - _Geoffrey Critzer_, Mar 21 2010
%e A055134 Triangle T(n,k) begins:
%e A055134        1;
%e A055134        0,      1;
%e A055134        1,      2,      1;
%e A055134        8,     12,      6,     1;
%e A055134       81,    108,     54,    12,    1;
%e A055134     1024,   1280,    640,   160,   20,   1;
%e A055134    15625,  18750,   9375,  2500,  375,  30,  1;
%e A055134   279936, 326592, 163296, 45360, 7560, 756, 42, 1;
%e A055134   ...
%t A055134 Clear[B] B[0] = {{x, y}, {t*y, x}}; B[n_] := B[n] = B[n - 1].B[0]; Table[Det[B[n]] /. x -> Sqrt[z] /. y -> 1 /. t -> n, {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[B[n]] /. x -> Sqrt[z] /. y ->1 /. t -> n, z], {n, 0, 10}]]; Flatten[a] (* _Roger L. Bagula_, Apr 09 2008 *)
%t A055134 row[n_] := CoefficientList[(x + n - 1)^n + O[x]^(n+1), x];
%t A055134 Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 13 2017, after _Geoffrey Critzer_ *)
%t A055134 Join[{1, 0, 1}, Table[Binomial[n, k]*(n - 1)^(n - k), {n, 2, 49}, {k, 0, n}]] // Flatten (* _G. C. Greubel_, Nov 14 2017 *)
%o A055134 (PARI) for(n=0,15, for(k=0,n, print1(if(n==0 && k==0, 1, if(n==1 && k==0, 0, if(n==1 && k==1, 1, binomial(n,k)*(n-1)^(n-k)))), ", "))) \\ _G. C. Greubel_, Nov 14 2017
%Y A055134 Columns k=0-2 give: A065440, A055897, A081132(n-2) for n>=2.
%Y A055134 Row sums give A000312.
%Y A055134 Cf. A055135, A055136.
%K A055134 nonn,tabl
%O A055134 0,5
%A A055134 _Christian G. Bower_, Apr 25 2000