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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).

%I A357078 #17 Sep 15 2022 14:44:00
%S A357078 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 A357078 2560,688,156,32,10,0,1,0,22368,5376,1080,216,40,12,0,1,0,216224,
%U A357078 46816,8456,1504,280,48,14,0,1,0,2291456,450240,73440,11808,1960,348,56,16,0,1
%N 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).
%C A357078 The partition transform (also called De Moivre polynomials by Cormac O'Sullivan) is defined in the program section as a Sage script.
%C A357078 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 A357078 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%e A357078 Triangle T(n, k) starts:                         [Row sums]
%e A357078 [0] 1;                                               [1]
%e A357078 [1] 0,      1;                                       [1]
%e A357078 [2] 0,      0,     1;                                [1]
%e A357078 [3] 0,      2,     0,    1;                          [3]
%e A357078 [4] 0,      8,     4,    0,    1;                    [13]
%e A357078 [5] 0,     48,    16,    6,    0,   1;               [71]
%e A357078 [6] 0,    328,   100,   24,    8,   0,  1;           [461]
%e A357078 [7] 0,   2560,   688,  156,   32,  10,  0,  1;       [3447]
%e A357078 [8] 0,  22368,  5376, 1080,  216,  40, 12,  0, 1;    [29093]
%e A357078 [9] 0, 216224, 46816, 8456, 1504, 280, 48, 14, 0, 1; [273343]
%o A357078 (SageMath)
%o A357078 from functools import cache
%o A357078 def PartTrans(dim, f):
%o A357078     X = var(['x' + str(i) for i in range(dim + 1)])
%o A357078     @cache
%o A357078     def PCoeffs(n: int, k: int):
%o A357078         R = PolynomialRing(ZZ, X[1: n - k + 2], n - k + 1, order='lex')
%o A357078         if k == 0: return R(k^n)
%o A357078         return R(sum(PCoeffs(n - j, k - 1) * f(j)
%o A357078                      for j in range(1, n - k + 2)))
%o A357078     return [[PCoeffs(n, k) for k in range(n + 1)] for n in range(dim)]
%o A357078 def A357078_triangle(dim):
%o A357078     A = ZZ[['t']]; g = A([0] + [factorial(n) for n in range(1, 30)]).O(dim+2)
%o A357078     return PartTrans(dim, lambda n: list(g / (1 + 2 * g))[n])
%o A357078 A357078_triangle(9)
%Y A357078 Cf. A355488, A003319, A059438, A356265, A357079, A269941.
%K A357078 nonn,tabl
%O A357078 0,8
%A A357078 _Peter Luschny_, Sep 10 2022