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.

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, 15, 9, 125, 132, 96, 85, 28, 17, 10, 281, 212, 119, 100, 32, 19, 11, 351, 469, 267, 142, 115, 36, 21, 12, 307, 642, 644, 322, 165, 130, 40, 23, 13

Offset: 0

Author

Alois P. Heinz, Nov 30 2015

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
00 :  1;
01 :     1;
02 :        2;
03 :        3,  3;
04 :            5,   4;
05 :            8,   7,   5;
06 :           16,  12,   9,   6;
07 :                40,  16,  11,   7;
08 :                34,  55,  20,  13,   8;
09 :                50,  73,  70,  24,  15,   9;
10 :               125, 132,  96,  85,  28,  17, 10;
11 :                    281, 212, 119, 100,  32, 19, 11;
12 :                    351, 469, 267, 142, 115, 36, 21, 12;
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• R. Stanley, Parking Functions, 2011

Crossrefs

Row sums give A265016.

Column sums give A265130.

Cf. A000217, A000272, A002024, A265019 (the same read by columns).

Programs

Maple

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
g:= (n, i, l)-> `if`(i*(i+1)/2n, 0, g(n-i, i-1, [i, l[]])))):
T:= n->(f->seq(coeff(f, x, i), i=ldegree(f)..degree(f)))(g(n$2, [])):
seq(T(n), n=0..20);

Formula

T(A000217(n),n) = A000272(n+1).