A318144 T(n, k) = (-1)^k*k!*P(n, k), where P(n, k) is the number of partitions of n of length k. Triangle read by rows, 0 <= k <= n.

1, 0, -1, 0, -1, 2, 0, -1, 2, -6, 0, -1, 4, -6, 24, 0, -1, 4, -12, 24, -120, 0, -1, 6, -18, 48, -120, 720, 0, -1, 6, -24, 72, -240, 720, -5040, 0, -1, 8, -30, 120, -360, 1440, -5040, 40320, 0, -1, 8, -42, 144, -600, 2160, -10080, 40320, -362880

Offset: 0

Author

Peter Luschny, Aug 20 2018

Keywords

• sign
• tabl

Examples

			[0] [1],
[1] [0, -1],
[2] [0, -1, 2],
[3] [0, -1, 2,  -6],
[4] [0, -1, 4,  -6,  24],
[5] [0, -1, 4, -12,  24, -120],
[6] [0, -1, 6, -18,  48, -120,  720],
[7] [0, -1, 6, -24,  72, -240,  720,  -5040],
[8] [0, -1, 8, -30, 120, -360, 1440,  -5040, 40320],
[9] [0, -1, 8, -42, 144, -600, 2160, -10080, 40320, -362880]
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened (first 45 rows from Peter Luschny)

Crossrefs

Row sums are A260845, absolute row sums are A101880.

Cf. A008284, A072233, A178803.

Programs

Magma

/* As triangle: */
[[(-1)^k*#Partitions(n,k)*Factorial(k): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Aug 20 2018

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1,
      b(n, i-1), 0)+expand(b(n-i, min(n-i, i))*x))
    end:
T:= n-> (p-> seq(i!*coeff(p, x, i)*(-1)^i, i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 18 2019

Mathematica

t[n_, k_] := (-1)^k  k! (IntegerPartitions[n, {k}] // Length);
Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > 1,
     b[n, i - 1], 0] + Expand[b[n - i, Min[n - i, i]]*x]];
T[n_] := Function[p, Table[i!*Coefficient[p, x, i]*(-1)^i, {i, 0, n}]][ b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Jun 07 2021, after Alois P. Heinz *)

Sage

from sage.combinat.partition import number_of_partitions_length
def A318144row(n):
    return [(-1)^k*number_of_partitions_length(n, k)*factorial(k) for k in (0..n)]
for n in (0..9): print(A318144row(n))