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A265019 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), k>=0, k<=n<=k*(k+1)/2, read by columns.

%I A265019 #22 Feb 11 2017 09:51:14
%S A265019 1,1,2,3,3,5,8,16,4,7,12,40,34,50,125,5,9,16,55,73,132,281,351,307,
%T A265019 432,1296,6,11,20,70,96,212,469,642,1020,1361,3294,3305,3910,3506,
%U A265019 4802,16807,7,13,24,85,119,267,644,959,1567,2686,5570,7109,11890,13234
%N A265019 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), k>=0, k<=n<=k*(k+1)/2, read by columns.
%H A265019 Alois P. Heinz, <a href="/A265019/b265019.txt">Columns k = 0..22, flattened</a>
%H A265019 R. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/parking.pdf">Parking Functions</a>, 2011.
%F A265019 T(A000217(n),n) = A000272(n+1).
%e A265019 Triangle T(n,k) begins:
%e A265019 00 :  1;
%e A265019 01 :     1;
%e A265019 02 :        2;
%e A265019 03 :        3,  3;
%e A265019 04 :            5,   4;
%e A265019 05 :            8,   7,   5;
%e A265019 06 :           16,  12,   9,   6;
%e A265019 07 :                40,  16,  11,   7;
%e A265019 08 :                34,  55,  20,  13,   8;
%e A265019 09 :                50,  73,  70,  24,  15,   9;
%e A265019 10 :               125, 132,  96,  85,  28,  17, 10;
%e A265019 11 :                    281, 212, 119, 100,  32, 19, 11;
%e A265019 12 :                    351, 469, 267, 142, 115, 36, 21, 12;
%p A265019 p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
%p A265019          -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
%p A265019 g:= (n, i, l)-> `if`(i*(i+1)/2<n, 0, `if`(n=0, p(l)*x^
%p A265019                 `if`(l=[], 0, l[-1]), g(n, i-1, l)+
%p A265019                 `if`(i>n, 0, g(n-i, i-1, [i, l[]])))):
%p A265019 b:= proc(n) option remember; g(n$2, []) end:
%p A265019 T:= k-> seq(coeff(b(n), x, k), n=k..k*(k+1)/2):
%p A265019 seq(T(k), k=0..8);
%t A265019 p[l_] := With[{n = Length[l]}, n!*Det[Table[t = j-i+1; If[t<0, 0, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]]; g[n_, i_, l_] := g[n, i, l] = If[i*(i+1)/2<n, 0, If[n==0, p[l]*x^If[l=={}, 0, l[[-1]]], g[n, i-1, l] + If[i>n, 0, g[n-i, i-1, Join[{i}, l]]]]]; b[n_] := b[n] = g[n, n, {}]; T[0] = {1}; T[k_] := Table[Coefficient[b[n], x, k], {n, k, k*(k+1)/2}]; Table[T[k], {k, 0, 8}] // Flatten (* _Jean-François Alcover_, Feb 11 2017, translated from Maple *)
%Y A265019 Row sums give A265016.
%Y A265019 Column sums give A265130.
%Y A265019 Cf. A000217, A000272, A265018 (the same read by rows).
%K A265019 nonn,tabf
%O A265019 0,3
%A A265019 _Alois P. Heinz_, Nov 30 2015