A357079 -id:A357079 - cp's OEIS frontend

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

Peter Luschny, Sep 10 2022

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.

			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]

Peter Luschny, The P-transform.

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

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)

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath