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.
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 10 2022
STATUS
approved