

A055137


Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows.


6



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
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OFFSET

0,7


COMMENTS

The nth row consists of coefficients of the characteristic polynomial of the adjacency matrix of the complete ngraph.
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 A132382the associated matrix for rB(0,1). The e.g.f. for the row polynomials is exp(x*t) * exp(x) *(1x). T(n,k) = Binomial(n,k)* s(nk) where s = (1,0,1,2,3,...) with an e.g.f. of exp(x)*(1x) 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 integroderivatives 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(nk) where s(n)= n, i.e., multiply the nth 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, December 3, 2005
M. Bhargava, K. Kedlaya, and L. Ng, Solutions to the 66th William Lowell Putnam Mathematical Competition Saturday, December 3, 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.: (xn+1)*(x+1)^(n1) = Sum_T(n, k)x^k.
T(n, k) = (1n+k)*C(n, k).
kth column has o.g.f. x^k(1(k+2)x)/(1x)^(k+2). kth row gives coefficients of (xk)(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)


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_] := (kn+1)*Binomial[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Nov 29 2013, after Pari *)


PROG

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


CROSSREFS

Cf. A005563, A005564 (absolute values of columns 1, 2).
Sequence in context: A191588 A106450 A255961 * A143325 A128888 A285320
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



