A357078 Triangle read by rows. The partition transform of A355488, which are the alternating row sums of the number of permutations of [n] with k components (A059438).

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 8, 4, 0, 1, 0, 48, 16, 6, 0, 1, 0, 328, 100, 24, 8, 0, 1, 0, 2560, 688, 156, 32, 10, 0, 1, 0, 22368, 5376, 1080, 216, 40, 12, 0, 1, 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1, 0, 2291456, 450240, 73440, 11808, 1960, 348, 56, 16, 0, 1

Offset

0,8

Comments

The partition transform (also called De Moivre polynomials by Cormac O'Sullivan) is defined in the program section as a Sage script.

The triangle represents a refinement of the number of irreducible permutations, A003319. Together with the refinement of the number of reducible permutations A356265 the triangle sums to the refinement of the factorial numbers given in A357079.

Links

Table of n, a(n) for n=0..65.

Peter Luschny, The P-transform.

Example

Triangle T(n, k) starts:                         [Row sums]
[0] 1;                                               [1]
[1] 0,      1;                                       [1]
[2] 0,      0,     1;                                [1]
[3] 0,      2,     0,    1;                          [3]
[4] 0,      8,     4,    0,    1;                    [13]
[5] 0,     48,    16,    6,    0,   1;               [71]
[6] 0,    328,   100,   24,    8,   0,  1;           [461]
[7] 0,   2560,   688,  156,   32,  10,  0,  1;       [3447]
[8] 0,  22368,  5376, 1080,  216,  40, 12,  0, 1;    [29093]
[9] 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1; [273343]

Prog

(SageMath)
from functools import cache
def PartTrans(dim, f):
    X = var(['x' + str(i) for i in range(dim + 1)])
    @cache
    def PCoeffs(n: int, k: int):
        R = PolynomialRing(ZZ, X[1: n - k + 2], n - k + 1, order='lex')
        if k == 0: return R(k^n)
        return R(sum(PCoeffs(n - j, k - 1) * f(j)
                     for j in range(1, n - k + 2)))
    return [[PCoeffs(n, k) for k in range(n + 1)] for n in range(dim)]
def A357078_triangle(dim):
    A = ZZ[['t']]; g = A([0] + [factorial(n) for n in range(1, 30)]).O(dim+2)
    return PartTrans(dim, lambda n: list(g / (1 + 2 * g))[n])
A357078_triangle(9)

Crossrefs

Cf. A355488, A003319, A059438, A356265, A357079, A269941.

Sequence in context: A049218 A212358 A344178 * A154469 A344584 A037273

Adjacent sequences: A357075 A357076 A357077 * A357079 A357080 A357081

Keyword

nonn,tabl

Author

Peter Luschny, Sep 10 2022

Status

approved