login
Triangle read by rows: T(n,k) = coefficient of [x^(n-k)] in the expansion of the polynomial (x+n)^n.
1

%I #35 Jan 10 2023 01:27:00

%S 1,1,1,1,4,4,1,9,27,27,1,16,96,256,256,1,25,250,1250,3125,3125,1,36,

%T 540,4320,19440,46656,46656,1,49,1029,12005,84035,352947,823543,

%U 823543,1,64,1792,28672,286720,1835008,7340032,16777216,16777216,1,81,2916,61236,826686,7440174,44641044

%N Triangle read by rows: T(n,k) = coefficient of [x^(n-k)] in the expansion of the polynomial (x+n)^n.

%C T(n,k) is the number of partial functions f on an n-set such that |dom(f)| = k. - _Geoffrey Critzer_, Jan 09 2023

%D O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009.

%F T(n,k) = binomial(n,k)*n^k.

%F E.g.f.: 1/(1-t(y*x))*exp(x*exp(t(y*x))) where t(z) is the e.g.f. for A000169. - _Geoffrey Critzer_, Jan 09 2023

%e n = 0, x^ 0 = 1;

%e n = 1, (x+1)^1 = x + 1, so 1, 1;

%e n = 2, (x+2)^2 = x^2 + 4x + 4, so 1, 4, 4;

%e n = 3, (x+3)^3 = x^3 + 9x^2 + 27x + 27, so 1, 9, 27, 27.

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 4, 4;

%e 1, 9, 27, 27;

%e 1, 16, 96, 256, 256;

%e 1, 25, 250, 1250, 3125, 3125;

%e ...

%p A243594 := proc(n,k)

%p expand( (x+n)^n) ;

%p coeftayl(%,x=0,n-k) ;

%p end proc:

%p for n from 0 to 12 do

%p for k from 0 to n do

%p printf("%d,",A243594(n,k)) ;

%p end do:

%p end do: # _R. J. Mathar_, Jun 10 2014

%t nn = 6; t[x_] := Sum[n^(n - 1) x^n/n!, {n, 1, nn}];Map[Select[#, # > 0 &] &,

%t Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t[ y x])]] Exp[x Exp[ t[y x]]], {x, 0,nn}], {x, y}]] // Grid (* _Geoffrey Critzer_, Jan 09 2023 *)

%o (PARI) row(n) = {my(p = (1+n*'x)^n); for (k=0, poldegree(p), print1(polcoeff(p, k), ", ")); } \\ _Michel Marcus_, Jun 10 2014

%Y Cf. A071207 (row reversed).

%Y Row sums give A000169.

%K nonn,tabl

%O 0,5

%A _Annette Kim_, Jun 07 2014