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A269941
Triangle read by rows, the coefficients of the partial P-polynomials.
16
1, 0, -1, 0, -1, 1, 0, -1, 2, -1, 0, -1, 1, 2, -3, 1, 0, -1, 2, 2, -3, -3, 4, -1, 0, -1, 1, 2, 2, -1, -6, -3, 6, 4, -5, 1, 0, -1, 2, 2, 2, -3, -3, -6, -3, 4, 12, 4, -10, -5, 6, -1, 0, -1, 1, 2, 2, 2, -3, -3, -6, -6, -3, 1, 12, 6, 12, 4, -10, -20, -5, 15, 6, -7, 1
OFFSET
0,9
COMMENTS
For the definition of the partial P-polynomials see the link 'P-transform'. The triangle of coefficients of the inverse partial P-polynomials is A269942.
LINKS
Peter Luschny, The P-transform, 2016.
Peter Luschny, The Partition Transform, A SageMath Jupyter Notebook, GitHub, 2016/2022.
Marko Riedel, Answer to Question 4943578, Mathematics Stack Exchange, 2024.
Peter Taylor, Answer to Question 474483, MathOverflow, 2024.
EXAMPLE
[[1]],
[[0], [-1]],
[[0], [-1], [1]],
[[0], [-1], [2], [-1]],
[[0], [-1], [1, 2], [-3], [1]],
[[0], [-1], [2, 2], [-3, -3], [4], [-1]],
[[0], [-1], [1, 2, 2], [-1, -6, -3], [6, 4], [-5], [1]],
[[0], [-1], [2, 2, 2], [-3, -3, -6, -3], [4, 12, 4], [-10, -5], [6], [-1]]
Replacing the sublists by their sums reduces the triangle to a signed version of the triangle A097805.
MAPLE
PTrans := proc(n, f, nrm:=NULL) local q, p, r, R;
if n = 0 then return [1] fi; R := [seq(0, j=0..n)];
for q in combinat:-partition(n) do
p := [op(ListTools:-Reverse(q)), 0]; r := p[1]+1;
mul(binomial(p[j], p[j+1])*f(j)^p[j], j=1..nops(q));
R[r] := R[r]-(-1)^r*% od;
if nrm = NULL then R else [seq(nrm(n, k)*R[k+1], k=0..n)] fi end:
A269941_row := n -> seq(coeffs(p), p in PTrans(n, n -> x[n])):
seq(lprint(A269941_row(n)), n=0..8);
PROG
(Sage)
def PtransMatrix(dim, f, norm = None, inverse = False, reduced = False):
i = 1; F = [1]
if reduced:
while i <= dim: F.append(f(i)); i += 1
else:
while i <= dim: F.append(F[i-1]*f(i)); i += 1
C = [[0 for k in range(m+1)] for m in range(dim)]
C[0][0] = 1
if inverse:
for m in (1..dim-1):
C[m][m] = -C[m-1][m-1]/F[1]
for k in range(m-1, 0, -1):
C[m][k] = -(C[m-1][k-1]+sum(F[i]*C[m][k+i-1]
for i in (2..m-k+1)))/F[1]
else:
for m in (1..dim-1):
C[m][m] = -C[m-1][m-1]*F[1]
for k in range(m-1, 0, -1):
C[m][k] = -sum(F[i]*C[m-i][k-1] for i in (1..m-k+1))
if norm == None: return C
for m in (1..dim-1):
for k in (1..m): C[m][k] *= norm(m, k)
return C
def PMultiCoefficients(dim, norm = None, inverse = False):
def coefficient(p):
if p <= 1: return [p]
return SR(p).fraction(ZZ).numerator().coefficients()
f = lambda n: var('x'+str(n))
P = PtransMatrix(dim, f, norm, inverse)
return [[coefficient(p) for p in L] for L in P]
print(flatten(PMultiCoefficients(9)))
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Peter Luschny, Mar 08 2016
STATUS
approved