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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

EXAMPLE

[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]

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

PROG

(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))

(MAGMA) /* As triangle: */

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

CROSSREFS

Row sums are A260845, absulute row sums are A101880.

Cf. A008284, A072233, A178803.

Sequence in context: A112570 A127755 A180662 * A260663 A241857 A300485

Adjacent sequences:  A318141 A318142 A318143 * A318145 A318146 A318147

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Aug 20 2018

STATUS

approved

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Last modified January 23 19:45 EST 2020. Contains 331175 sequences. (Running on oeis4.)