A357078 - OEIS

%I #17 Sep 15 2022 14:44:00

%S 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,

%T 2560,688,156,32,10,0,1,0,22368,5376,1080,216,40,12,0,1,0,216224,

%U 46816,8456,1504,280,48,14,0,1,0,2291456,450240,73440,11808,1960,348,56,16,0,1

%N 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).

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

%C 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.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.

%e Triangle T(n, k) starts:                         [Row sums]

%e [0] 1;                                               [1]

%e [1] 0,      1;                                       [1]

%e [2] 0,      0,     1;                                [1]

%e [3] 0,      2,     0,    1;                          [3]

%e [4] 0,      8,     4,    0,    1;                    [13]

%e [5] 0,     48,    16,    6,    0,   1;               [71]

%e [6] 0,    328,   100,   24,    8,   0,  1;           [461]

%e [7] 0,   2560,   688,  156,   32,  10,  0,  1;       [3447]

%e [8] 0,  22368,  5376, 1080,  216,  40, 12,  0, 1;    [29093]

%e [9] 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1; [273343]

%o (SageMath)

%o from functools import cache

%o def PartTrans(dim, f):

%o     X = var(['x' + str(i) for i in range(dim + 1)])

%o     @cache

%o     def PCoeffs(n: int, k: int):

%o         R = PolynomialRing(ZZ, X[1: n - k + 2], n - k + 1, order='lex')

%o         if k == 0: return R(k^n)

%o         return R(sum(PCoeffs(n - j, k - 1) * f(j)

%o                      for j in range(1, n - k + 2)))

%o     return [[PCoeffs(n, k) for k in range(n + 1)] for n in range(dim)]

%o def A357078_triangle(dim):

%o     A = ZZ[['t']]; g = A([0] + [factorial(n) for n in range(1, 30)]).O(dim+2)

%o     return PartTrans(dim, lambda n: list(g / (1 + 2 * g))[n])

%o A357078_triangle(9)

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

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Sep 10 2022