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A265018 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.

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