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A055137 Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows. 9
1, 0, 1, -1, 0, 1, -2, -3, 0, 1, -3, -8, -6, 0, 1, -4, -15, -20, -10, 0, 1, -5, -24, -45, -40, -15, 0, 1, -6, -35, -84, -105, -70, -21, 0, 1, -7, -48, -140, -224, -210, -112, -28, 0, 1, -8, -63, -216, -420, -504, -378, -168, -36, 0, 1, -9, -80, -315, -720 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

The n-th row consists of coefficients of the characteristic polynomial of the adjacency matrix of the complete n-graph.

Triangle of coefficients of det(M(n)) where M(n) is the n X n matrix m(i,j)=x if i=j, m(i,j)=i/j otherwise. - Benoit Cloitre, Feb 01 2003

T is an example of the group of matrices outlined in the table in A132382--the associated matrix for rB(0,1). The e.g.f. for the row polynomials is exp(x*t) * exp(x) *(1-x). T(n,k) = Binomial(n,k)* s(n-k) where s = (1,0,-1,-2,-3,...) with an e.g.f. of exp(x)*(1-x) which is the reciprocal of the e.g.f. of A000166. - Tom Copeland, Sep 10 2008

Row sums are: {1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}. - Roger L. Bagula, Feb 20 2009

T is related to an operational calculus connecting an infinitesimal generator for fractional integro-derivatives with the values of the Riemann zeta function at positive integers (see MathOverflow links). - Tom Copeland, Nov 02 2012

The submatrix below the null subdiagonal is signed and row reversed A127717. The submatrix below the diagonal is A074909(n,k)s(n-k) where s(n)= -n, i.e., multiply the n-th diagonal by -n. A074909 and its reverse A135278 have several combinatorial interpretations. - Tom Copeland, Nov 04 2012

REFERENCES

Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. p. 17.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p.184 problem 3.

LINKS

Table of n, a(n) for n=0..58.

Problem B6, The 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005

M. Bhargava, K. Kedlaya, and L. Ng, Solutions to the 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005

T. Copeland, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

T. Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

FORMULA

G.f.: (x-n+1)*(x+1)^(n-1) = Sum_(k=0..n) T(n,k) x^k.

T(n, k) = (1-n+k)*binomial(n, k).

k-th column has o.g.f. x^k(1-(k+2)x)/(1-x)^(k+2). k-th row gives coefficients of (x-k)(x+1)^k. - Paul Barry, Jan 25 2004

T(n,k) = Coefficientslist[Det[Table[If[i == j, x, 1], {i, 1, n}, {k, 1, n}],x]. - Roger L. Bagula, Feb 20 2009

From Peter Bala, Aug 08 2011: (Start)

Given a permutation p belonging to the symmetric group S_n, let fix(p) be the number of fixed points of p and sign(p) its parity. The row polynomials R(n,x) have a combinatorial interpretation as R(n,x) = (-1)^n*Sum_{permutations p in S_n} sign(p)*(-x)^(fix(p)). An example is given below.

Note: The polynomials P(n,x) = Sum_{permutations p in S_n} x^(fix(p)) are the row polynomials of the rencontres numbers A008290. The integral results Integral_{x = 0..n} R(n,x) dx = n/(n+1) = Integral_{x = 0..-1} R(n,x) dx lead to the identities Sum_{p in S_n} sign(p)*(-n)^(1 + fix(p))/(1 + fix(p)) = (-1)^(n+1)*n/(n+1) = Sum_{p in S_n} sign(p)/(1 + fix(p)). The latter equality was Problem B6 in the 66th William Lowell Putnam Mathematical Competition 2005. (End)

From Tom Copeland, Jul 26 2017: (Start)

The e.g.f. in Copeland's 2008 comment implies this entry is an Appell sequence of polynomials P(n,x) with lowering and raising operators L = d/dx and R = x + d/dL log[exp(L)(1-L)] = x+1 - 1/(1-L) = x - L - L^2 - ... such that L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).

P(n,x) = (1-L) exp(L) x^n = (1-L) (x+1)^n = (x+1)^n - n (x+1)^(n-1) = (x+1-n)(x+1)^(n-1) = (x+s.)^n umbrally, where (s.)^n = s_n = P(n,0).

The formalism of A133314 applies to the pair of entries A008290 and A055137.

The polynomials of this pair P_n(x) and Q_n(x) are umbral compositional inverses; i.e., P_n(Q.(x)) = x^n = Q_n(P.(x)), where, e.g., (Q.(x))^n = Q_n(x).

The exponential infinitesimal generator (infinigen) of this entry is the negated infinigen of A008290, the matrix (M) noted by Bala, related to A238363. Then e^M = [the lower triangular A008290], and e^(-M) = [the lower triangular A055137]. For more on the infinigens, see A238385. (End)

From the row g.f.s corresponding to Bagula's matrix example below, the n-th row polynomial has a zero of multiplicity n-1 at x = 1 and a zero at x = -n+1. Since this is an Appell sequence dP_n(x)/dx = n P_{n-1}(x), the critical points of P_n(x) have the same abscissas as the zeros of P_{n-1}(x); therefore, x = 1 is an inflection point for the polynomials of degree > 2 with P_n(1) = 0, and the one local extremum of P_n has the abscissa x = -n + 2 with the value (-n+1)^{n-1}, signed values of A000312. - Tom Copeland, Nov 15 2019

EXAMPLE

1; 0,1; -1,0,1; -2,-3,0,1; -3,-8,-6,0,1; ...

(Bagula's matrix has a different sign convention from the list.)

From Roger L. Bagula, Feb 20 2009: (Start)

  { 1},

  { 0,   1},

  {-1,   0,    1},

  { 2,  -3,    0,    1},

  {-3,   8,   -6,    0,     1},

  { 4, -15,   20,  -10,     0,    1},

  {-5,  24,  -45,   40,   -15,    0,    1},

  { 6, -35,   84, -105,    70,  -21,    0,   1},

  {-7,  48, -140,  224,  -210,  112,  -28,   0,   1},

  { 8, -63,  216, -420,   504, -378,  168, -36,   0, 1},

  {-9,  80, -315,  720, -1050, 1008, -630, 240, -45, 0, 1}

(End)

R(3,x) = (-1)^3*Sum_{permutations p in S_3} sign(p)*(-x)^(fix(p)).

    p   | fix(p) | sign(p) | (-1)^3*sign(p)*(-x)^fix(p)

========+========+=========+===========================

  (123) |    3   |    +1   |      x^3

  (132) |    1   |    -1   |       -x

  (213) |    1   |    -1   |       -x

  (231) |    0   |    +1   |       -1

  (312) |    0   |    +1   |       -1

  (321) |    1   |    -1   |       -x

========+========+=========+===========================

                           | R(3,x) = x^3 - 3*x - 2

- Peter Bala, Aug 08 2011

MATHEMATICA

M[n_] := Table[If[i == j, x, 1], {i, 1, n}, {j, 1, n}]; a = Join[{{1}}, Flatten[Table[CoefficientList[Det[M[n]], x], {n, 1, 10}]]] (* Roger L. Bagula, Feb 20 2009 *)

t[n_, k_] := (k-n+1)*Binomial[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Nov 29 2013, after Pari *)

PROG

(PARI) T(n, k)=(1-n+k)*if(k<0 || k>n, 0, n!/k!/(n-k)!)

CROSSREFS

Cf. A005563, A005564 (absolute values of columns 1, 2).

Cf. A008290, A133314, A238363, A238385.

Cf. A000312.

Sequence in context: A106450 A255961 A297328 * A143325 A307910 A128888

Adjacent sequences:  A055134 A055135 A055136 * A055138 A055139 A055140

KEYWORD

sign,tabl

AUTHOR

Christian G. Bower, Apr 25 2000

EXTENSIONS

Additional comments from Michael Somos, Jul 04 2002

STATUS

approved

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Last modified May 30 08:43 EDT 2020. Contains 334712 sequences. (Running on oeis4.)