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A269941 Triangle read by rows, the coefficients of the partial P-polynomials.

%I A269941 #33 Aug 26 2024 04:33:45
%S A269941 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,
%T A269941 -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,
%U A269941 2,2,-3,-3,-6,-6,-3,1,12,6,12,4,-10,-20,-5,15,6,-7,1
%N A269941 Triangle read by rows, the coefficients of the partial P-polynomials.
%C A269941 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.
%H A269941 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>, 2016.
%H A269941 Peter Luschny, <a href="https://github.com/PeterLuschny/PartitionTransform/blob/main/PartitionTransform.ipynb">The Partition Transform</a>, A SageMath Jupyter Notebook, GitHub, 2016/2022.
%H A269941 Marko Riedel, <a href="https://math.stackexchange.com/q/4943578">Answer to Question 4943578</a>, Mathematics Stack Exchange, 2024.
%H A269941 Peter Taylor, <a href="https://mathoverflow.net/q/474504">Answer to Question 474483</a>, MathOverflow, 2024.
%e A269941 [[1]],
%e A269941 [[0], [-1]],
%e A269941 [[0], [-1], [1]],
%e A269941 [[0], [-1], [2], [-1]],
%e A269941 [[0], [-1], [1, 2], [-3], [1]],
%e A269941 [[0], [-1], [2, 2], [-3, -3], [4], [-1]],
%e A269941 [[0], [-1], [1, 2, 2], [-1, -6, -3], [6, 4], [-5], [1]],
%e A269941 [[0], [-1], [2, 2, 2], [-3, -3, -6, -3], [4, 12, 4], [-10, -5], [6], [-1]]
%e A269941 Replacing the sublists by their sums reduces the triangle to a signed version of the triangle A097805.
%p A269941 PTrans := proc(n, f, nrm:=NULL) local q, p, r, R;
%p A269941 if n = 0 then return [1] fi; R := [seq(0,j=0..n)];
%p A269941 for q in combinat:-partition(n) do
%p A269941    p := [op(ListTools:-Reverse(q)),0]; r := p[1]+1;
%p A269941    mul(binomial(p[j], p[j+1])*f(j)^p[j], j=1..nops(q));
%p A269941    R[r] := R[r]-(-1)^r*% od;
%p A269941 if nrm = NULL then R else [seq(nrm(n,k)*R[k+1],k=0..n)] fi end:
%p A269941 A269941_row := n -> seq(coeffs(p), p in PTrans(n, n -> x[n])):
%p A269941 seq(lprint(A269941_row(n)), n=0..8);
%o A269941 (Sage)
%o A269941 def PtransMatrix(dim, f, norm = None, inverse = False, reduced = False):
%o A269941     i = 1; F = [1]
%o A269941     if reduced:
%o A269941         while i <= dim: F.append(f(i)); i += 1
%o A269941     else:
%o A269941         while i <= dim: F.append(F[i-1]*f(i)); i += 1
%o A269941     C = [[0 for k in range(m+1)] for m in range(dim)]
%o A269941     C[0][0] = 1
%o A269941     if inverse:
%o A269941         for m in (1..dim-1):
%o A269941             C[m][m] = -C[m-1][m-1]/F[1]
%o A269941             for k in range(m-1, 0, -1):
%o A269941                 C[m][k] = -(C[m-1][k-1]+sum(F[i]*C[m][k+i-1]
%o A269941                           for i in (2..m-k+1)))/F[1]
%o A269941     else:
%o A269941         for m in (1..dim-1):
%o A269941             C[m][m] = -C[m-1][m-1]*F[1]
%o A269941             for k in range(m-1, 0, -1):
%o A269941                 C[m][k] = -sum(F[i]*C[m-i][k-1] for i in (1..m-k+1))
%o A269941     if norm == None: return C
%o A269941     for m in (1..dim-1):
%o A269941         for k in (1..m): C[m][k] *= norm(m,k)
%o A269941     return C
%o A269941 def PMultiCoefficients(dim, norm = None, inverse = False):
%o A269941     def coefficient(p):
%o A269941         if p <= 1: return [p]
%o A269941         return SR(p).fraction(ZZ).numerator().coefficients()
%o A269941     f = lambda n: var('x'+str(n))
%o A269941     P = PtransMatrix(dim, f, norm, inverse)
%o A269941     return [[coefficient(p) for p in L] for L in P]
%o A269941 print(flatten(PMultiCoefficients(9)))
%Y A269941 Cf. A097805, A268441, A268442, A269942.
%K A269941 sign,tabf
%O A269941 0,9
%A A269941 _Peter Luschny_, Mar 08 2016