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%I #90 Nov 16 2019 01:38:06
%S 1,0,1,-1,0,1,-2,-3,0,1,-3,-8,-6,0,1,-4,-15,-20,-10,0,1,-5,-24,-45,
%T -40,-15,0,1,-6,-35,-84,-105,-70,-21,0,1,-7,-48,-140,-224,-210,-112,
%U -28,0,1,-8,-63,-216,-420,-504,-378,-168,-36,0,1,-9,-80,-315,-720
%N Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows.
%C The n-th row consists of coefficients of the characteristic polynomial of the adjacency matrix of the complete n-graph.
%C 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
%C 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
%C Row sums are: {1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}. - _Roger L. Bagula_, Feb 20 2009
%C 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
%C 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
%D Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. p. 17.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p.184 problem 3.
%H Problem B6, <a href="http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2005.pdf">The 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005</a>
%H M. Bhargava, K. Kedlaya, and L. Ng, <a href="http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2005s.pdf">Solutions to the 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005</a>
%H T. Copeland, <a href="http://mathoverflow.net/questions/111770/cycling-through-the-zeta-garden-zeta-functions-for-graphs-cycle-index-polynomi">Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants</a>
%H T. Copeland, <a href="http://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials">Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus</a>
%F G.f.: (x-n+1)*(x+1)^(n-1) = Sum_(k=0..n) T(n,k) x^k.
%F T(n, k) = (1-n+k)*binomial(n, k).
%F 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
%F T(n,k) = Coefficientslist[Det[Table[If[i == j, x, 1], {i, 1, n}, {k, 1, n}],x]. - _Roger L. Bagula_, Feb 20 2009
%F From _Peter Bala_, Aug 08 2011: (Start)
%F 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.
%F 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)
%F From _Tom Copeland_, Jul 26 2017: (Start)
%F 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).
%F 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).
%F The formalism of A133314 applies to the pair of entries A008290 and A055137.
%F 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).
%F 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)
%F 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
%e 1; 0,1; -1,0,1; -2,-3,0,1; -3,-8,-6,0,1; ...
%e (Bagula's matrix has a different sign convention from the list.)
%e From _Roger L. Bagula_, Feb 20 2009: (Start)
%e { 1},
%e { 0, 1},
%e {-1, 0, 1},
%e { 2, -3, 0, 1},
%e {-3, 8, -6, 0, 1},
%e { 4, -15, 20, -10, 0, 1},
%e {-5, 24, -45, 40, -15, 0, 1},
%e { 6, -35, 84, -105, 70, -21, 0, 1},
%e {-7, 48, -140, 224, -210, 112, -28, 0, 1},
%e { 8, -63, 216, -420, 504, -378, 168, -36, 0, 1},
%e {-9, 80, -315, 720, -1050, 1008, -630, 240, -45, 0, 1}
%e (End)
%e R(3,x) = (-1)^3*Sum_{permutations p in S_3} sign(p)*(-x)^(fix(p)).
%e p | fix(p) | sign(p) | (-1)^3*sign(p)*(-x)^fix(p)
%e ========+========+=========+===========================
%e (123) | 3 | +1 | x^3
%e (132) | 1 | -1 | -x
%e (213) | 1 | -1 | -x
%e (231) | 0 | +1 | -1
%e (312) | 0 | +1 | -1
%e (321) | 1 | -1 | -x
%e ========+========+=========+===========================
%e | R(3,x) = x^3 - 3*x - 2
%e - _Peter Bala_, Aug 08 2011
%t 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 t[n_, k_] := (k-n+1)*Binomial[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2013, after Pari *)
%o (PARI) T(n,k)=(1-n+k)*if(k<0 || k>n,0,n!/k!/(n-k)!)
%Y Cf. A005563, A005564 (absolute values of columns 1, 2).
%Y Cf. A008290, A133314, A238363, A238385.
%Y Cf. A000312.
%K sign,tabl
%O 0,7
%A _Christian G. Bower_, Apr 25 2000
%E Additional comments from _Michael Somos_, Jul 04 2002
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