%I #17 Dec 04 2015 21:24:40
%S 1,1,2,3,3,5,4,8,7,5,16,12,9,6,40,16,11,7,34,55,20,13,8,50,73,70,24,
%T 15,9,125,132,96,85,28,17,10,281,212,119,100,32,19,11,351,469,267,142,
%U 115,36,21,12,307,642,644,322,165,130,40,23,13
%N Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with largest part k; triangle T(n,k), n>=0, floor(sqrt(2n)+1/2)<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A265018/b265018.txt">Rows n = 0..100, flattened</a>
%H R. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011
%F T(A000217(n),n) = A000272(n+1).
%e Triangle T(n,k) begins:
%e 00 : 1;
%e 01 : 1;
%e 02 : 2;
%e 03 : 3, 3;
%e 04 : 5, 4;
%e 05 : 8, 7, 5;
%e 06 : 16, 12, 9, 6;
%e 07 : 40, 16, 11, 7;
%e 08 : 34, 55, 20, 13, 8;
%e 09 : 50, 73, 70, 24, 15, 9;
%e 10 : 125, 132, 96, 85, 28, 17, 10;
%e 11 : 281, 212, 119, 100, 32, 19, 11;
%e 12 : 351, 469, 267, 142, 115, 36, 21, 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=ldegree(f)..degree(f)))(g(n$2, [])):
%p seq(T(n), n=0..20);
%Y Row sums give A265016.
%Y Column sums give A265130.
%Y Cf. A000217, A000272, A002024, A265019 (the same read by columns).
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Nov 30 2015