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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

Table of n, a(n) for n=0..51.

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.)