A268442 -id:A268442

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A269941

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

+10
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

(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..74.

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

Cf. A097805, A268441, A268442, A269942.

KEYWORD

sign,tabf

AUTHOR

Peter Luschny, Mar 08 2016

STATUS

approved

A268441

Triangle read by rows, the coefficients of the Bell polynomials.

+10
7

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 3, 4, 6, 1, 0, 1, 10, 5, 15, 10, 10, 1, 0, 1, 10, 15, 6, 15, 60, 15, 45, 20, 15, 1, 0, 1, 35, 21, 7, 105, 70, 105, 21, 105, 210, 35, 105, 35, 21, 1, 0, 1, 35, 56, 28, 8, 280, 210, 280, 168, 28, 105, 840, 280, 420, 56, 420, 560, 70, 210, 56, 28, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

The triangle of coefficients of the inverse Bell polynomials is A268442.

REFERENCES

L. Comtet, Advanced combinatorics, The art of finite and infinite expansions, 1974.

LINKS

Peter Luschny, First 26 rows, flattened

E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.

E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.

Peter Luschny, The Bell transform

FORMULA

E.g.f.: exp( Sum_{k>=1} x_{k}*t^k/k! ), monomials in negative lexicographic order.

EXAMPLE

[[1]]

[[0], [1]]

[[0], [1], [1]]

[[0], [1], [3], [1]]

[[0], [1], [3, 4], [6], [1]]

[[0], [1], [10, 5], [15, 10], [10], [1]]

[[0], [1], [10, 15, 6], [15, 60, 15], [45, 20], [15], [1]]

Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of second kind (A048993).

MATHEMATICA

BellCoeffs[n_, k_] := Module[{v, r},

v = Table[Subscript[x, j], {j, 1, n}]; (* list of variables *)

r = Table[Subscript[x, j]->1, {j, 1, n}]; (* evaluated at 1 *)

MonomialList[BellY[n, k, v], v, NegativeLexicographic] /. r];

A268441Row[n_] := Table[BellCoeffs[n, k], {k, 0, n}] // Flatten;

Do[Print[A268441Row[n]], {n, 0, 8}] (* Peter Luschny, Feb 08 2016 *)

max = 9; egf = Exp[Sum[x[k]*t^k/k!, {k, 1, max}]]; P = Table[n!* SeriesCoefficient[egf, {t, 0, n}], {n, 0, max-1}]; row[n_] := (s = Split[ Sort[{ Exponent[# /. x[_] -> x, x], #}& /@ (List @@ Expand[P[[n]]])], #1[[1]] == #2[[1]]&]; Join[{0}, #[[All, 2]]& /@ (s /. x[_] -> 1) // Flatten]); row[1] = {1}; Array[row, max] // Flatten (* Jean-François Alcover, Feb 08 2016 *)

PROG

(Sage)

import itertools

def A268441_row(n):

c = [bell_polynomial(n, k).coefficients() for k in (0..n)]

if n>0: c[0] = [0]

return list(itertools.chain(*c))

for n in range(9): print(A268441_row(n))

CROSSREFS

Cf. A048993, A268442.

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Feb 07 2016

STATUS

approved

A269942

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

+10
1

1, 0, -1, 0, -1, 1, 0, -2, 1, 2, -1, 0, -5, 5, -1, 5, -2, -3, 1, 0, -14, 21, -3, -6, 1, 14, -12, 2, -9, 3, 4, -1, 0, -42, 84, -28, -28, 7, 7, -1, 42, -56, 7, 14, -2, -28, 21, -3, 14, -4, -5, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

The triangle of coefficients of the partial P-polynomials is A269941. For the definition of the inverse partial P-polynomials see the link 'P-transform'.

LINKS

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

Peter Luschny, The P-transform.

EXAMPLE

[[1]],

[[0], [-1]],

[[0], [-1], [1]],

[[0], [-2, 1], [2], [-1]],

[[0], [-5, 5, -1], [5, -2], [-3], [1]],

[[0], [-14, 21, -3, -6, 1], [14, -12, 2], [-9, 3], [4], [-1]]],

[[0], [-42,84,-28,-28,7,7,-1],[42,-56,7,14,-2],[-28,21,-3],[14,-4],[-5],[1]]

Replacing the sublists by their sums reduces the triangle to a signed version of the triangle A097805. The column 1 of sublists is A111785 in a different order.

PROG

(Sage)

# For function PMultiCoefficients see A269941.

PMultiCoefficients(7, inverse = True)

CROSSREFS

Cf. A097805, A111785, A268441, A268442, A269941.

KEYWORD

sign,tabf

AUTHOR

Peter Luschny, Mar 08 2016

STATUS

approved

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