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%I #21 Aug 20 2021 04:27:06
%S 1,0,1,0,0,2,0,3,0,3,0,5,0,0,4,0,7,8,0,0,5,0,25,12,0,0,0,6,0,36,16,15,
%T 0,0,0,7,0,81,20,21,0,0,0,0,8,0,107,74,27,24,0,0,0,0,9,0,316,102,33,
%U 32,0,0,0,0,0,10,0,427,222,39,40,35,0,0,0,0,0,11
%N Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with smallest part k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A265017/b265017.txt">Rows n = 0..100, flattened</a>
%H Richard P. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011.
%e Triangle T(n,k) begins:
%e 00 : 1;
%e 01 : 0, 1;
%e 02 : 0, 0, 2;
%e 03 : 0, 3, 0, 3;
%e 04 : 0, 5, 0, 0, 4;
%e 05 : 0, 7, 8, 0, 0, 5;
%e 06 : 0, 25, 12, 0, 0, 0, 6;
%e 07 : 0, 36, 16, 15, 0, 0, 0, 7;
%e 08 : 0, 81, 20, 21, 0, 0, 0, 0, 8;
%e 09 : 0, 107, 74, 27, 24, 0, 0, 0, 0, 9;
%e 10 : 0, 316, 102, 33, 32, 0, 0, 0, 0, 0, 10;
%e 11 : 0, 427, 222, 39, 40, 35, 0, 0, 0, 0, 0, 11;
%e 12 : 0, 869, 286, 153, 48, 45, 0, 0, 0, 0, 0, 0, 12;
%p p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
%p -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
%p g:= (n, i, l)-> `if`(i*(i+1)/2<n, 0, `if`(n=0, p(l)*x^
%p `if`(l=[], 0, l[1]), g(n, i-1, l)+
%p `if`(i>n, 0, g(n-i, i-1, [i, l[]])))):
%p T:= n-> (f-> seq(coeff(f, x, i), i=0..n))(g(n$2, [])):
%p seq(T(n), n=0..16);
%t p[l_] := With[{n = Length[l]}, n!*Det[Table[Function[t,
%t If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]];
%t g[n_, i_, l_] := If[i(i+1)/2 < n, 0, If[n == 0, p[l]*x^
%t If[l == {}, 0, l[[1]]], g[n, i - 1, l] +
%t If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
%t T[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];
%t Table[T[n], {n, 0, 16}] // Flatten (* _Jean-François Alcover_, Aug 20 2021, after _Alois P. Heinz_ *)
%Y Row sums give A265016.
%Y Column k=0 gives A000007.
%Y Main diagonal gives A028310, first lower diagonal is A000004.
%Y T(2n+1,n) gives A005563.
%Y T(2n+2,n) gives A028347(n+2).
%Y T(2n+3,n) gives A028560.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Nov 30 2015
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