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 A269941 Triangle read by rows, the coefficients of the partial P-polynomials. 11
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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] PMultiCoefficients(8) CROSSREFS Cf. A097805, A268441, A268442, A269942. Sequence in context: A143841 A276719 A276837 * A035440 A029878 A182458 Adjacent sequences:  A269938 A269939 A269940 * A269942 A269943 A269944 KEYWORD sign,tabf AUTHOR Peter Luschny, Mar 08 2016 STATUS approved

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Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)