OFFSET
0,5
COMMENTS
Differs from A265020 first at T(5,2). See example.
LINKS
Alois P. Heinz, Rows n = 0..400, flattened
R. Stanley, Parking Functions, 2011
EXAMPLE
T(5,2) = 10: There are two partitions of 5 into 2 distinct parts: [2,3], [1,4]. Together they have 10 lambda-parking functions: [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [3,1], [3,2], [4,1]. Here [1,1], [1,2], [1,3], [2,1], [3,1] are induced by both partitions. But they are counted only once.
T(6,1) = 6: [1], [2], [3], [4], [5], [6].
T(6,2) = 14: [1,1], [1,2], [1,3], [1,4], [1,5], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [4,1], [4,2], [5,1].
T(6,3) = 16: [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,3,1], [3,1,1], [3,1,2], [3,2,1].
Triangle T(n,k) begins:
00 : 1;
01 : 0, 1;
02 : 0, 2;
03 : 0, 3, 3;
04 : 0, 4, 5;
05 : 0, 5, 10;
06 : 0, 6, 14, 16;
07 : 0, 7, 21, 25;
08 : 0, 8, 27, 43;
09 : 0, 9, 36, 74;
10 : 0, 10, 44, 107, 125;
11 : 0, 11, 55, 146, 189;
12 : 0, 12, 65, 207, 307;
13 : 0, 13, 78, 267, 471;
14 : 0, 14, 90, 342, 786;
15 : 0, 15, 105, 436, 1058, 1296;
16 : 0, 16, 119, 538, 1490, 1921;
MAPLE
b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!*x^p)+
`if`(n<t, 0, add(b(p+1, `if`(i=j, g+1, 1), n-max(j, t), j,
max(j, t)+1)/`if`(i=j, 1, g!), j=i..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
`if`(n=0, 1, b(0$2, n, 1$2))):
seq(T(n), n=0..25);
MATHEMATICA
b[p_, g_, n_, i_, t_] := b[p, g, n, i, t] = If[g==0, 0, p!/g!*x^p] + If[n<t, 0, Sum[b[p+1, If[i==j, g+1, 1], n-Max[j, t], j, Max[j, t]+1] / If[i==j, 1, g!], {j, i, n}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][If[n==0, 1, b[0, 0, n, 1, 1]]]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Feb 02 2017, translated from Maple *)
CROSSREFS
Row sums give A265202.
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Dec 04 2015
STATUS
approved