A355488 -id:A355488 - 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)

A355775 Expansion of 1 + f/(1 + 2*f), where f is the g.f. of the swinging factorial (A056040).

Original entry on oeis.org

1, 1, 0, 2, -10, 30, -84, 244, -722, 2126, -6252, 18364, -53988, 158668, -466408, 1370792, -4029154, 11842222, -34806972, 102303468, -300690348, 883782404, -2597604952, 7634834648, -22440207764, 65955900268, -193856647736, 569780485080, -1674690451976

Offset: 0

Peter Luschny, Jul 22 2022

sign

Cf. A056040, A355488.

# Traditional style:
u := 1 - 4*z^2: v := u + z: w := u^(3/2) - v: 1 + w/(w - v):
ser := series(%, z, 30): seq(coeff(%, z, n), n = 0..28);
# Alternative (Alois P. Heinz style, see A355488):
c := n -> n! / iquo(n, 2)!^2:
a := n -> (f -> coeff(series(1 + f/(1 + 2*f), x, n+1), x, n))(add(c(j)*x^j, j=1..n)): seq(a(n), n = 0..28);

c[n_] := n! / Quotient[n, 2]!^2;
a[n_] := With[{f = Sum[c[j]*x^j, {j, 1, n}]}, SeriesCoefficient[1 + f/(1 + 2*f), {x, 0, n}]];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Apr 17 2025, after Alois P. Heinz *)

def sw_factorial(n): return factorial(n) // factorial(n // 2)^2
A = QQ[['t']]
f = A([0] + [sw_factorial(n) for n in range(1, 29)]).O(29)
print(list(1 + f / (1 + 2 * f)))  # After F. Chapoton in A355488.

a(n) = [z^n] (1 + ((1 - 4*z^2)^(3/2) + 4*z^2 - z - 1) / ((1 - 4*z^2)^(3/2) + 2*(4*z^2 - z - 1))).

Let u = 1 - 4*z^2, v = u + z and w = u^(3/2) - v. Then the g.f. can be written as 1 + w/(w - v) and the g.f. of the swinging factorial 1 - w/(w + v).

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).

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