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%I A131758
%S A131758 1,0,1,1,1,2,4,14,10,6,15,83,157,89,24,56,424,1266,1724,826,120,185,
%T A131758 1887,8038,17642,19593,8287,720,204,4976,36226,126944,239576,234688,
%U A131758 90602,5040
%V A131758 1,0,1,-1,1,2,4,-14,10,6,-15,83,-157,89,24,56,-424,1266,-1724,826,120,
               -185,1887,-8038,
%W A131758 17642,-19593,8287,720,204,-4976,36226,-126944,239576,-234688,90602,5040
%N A131758 Coefficients of numerators of rational functions whose binomial transforms 
               give the normalized polylogarithms Li(-n,t)/ n!.
%C A131758 Coefficients may be generated from a modified Riordan array (MRA) formed 
               from Rgf(z,t) = [ t/(1+z) ]/{exp(-z/(1+z))-t} with each row of the 
               array acting to generate the succeeding polynomial P(n,t) from the 
               preceding n polynomials.
%C A131758 The MRA is constructed by appending an n! to the left of the n-th row 
               of the Riordan array A129652 and removing the unit diagonal.
%C A131758 The MRA is partially
%C A131758 1
%C A131758 1, 1
%C A131758 2, 3, 2
%C A131758 6, 13, 9, 3
%C A131758 24, 73, 52, 18, 4
%C A131758 120, 501, 365, 130, 30, 5
%C A131758 720, 4051, 3006, 1095, 260, 45, 6
%C A131758 For the MRA:
%C A131758 1) First column is the n!'s.
%C A131758 2) Second column is A000262.
%C A131758 Then, e.g., from the terms in the MRA
%C A131758 P(0,t) = 0!*(t-1)^0 = 1 from the n=0 row,
%C A131758 P(1,t) = 1!*(t-1)^1 + 1*P(0,t) = t from the n=1 row,
%C A131758 P(2,t) = 2!*(t-1)^2 + 3*P(0,t)*(t-1)^1 + 2*P(1,t)
%C A131758 P(3,t) = 3!*(t-1)^3 + 13*P(0,t)*(t-1)^2 + 9*P(1,t)*(t-1)^1 + 3*P(2,t)
%C A131758 Generating
%C A131758 P(0,t) = (1)
%C A131758 P(1,t) = (0, 1)
%C A131758 P(2,t) = (-1, 1, 2)
%C A131758 P(3,t) = (4, -14, 10, 6) = 4 + -14 t + 10 t^2 + 6 t^3
%C A131758 P(4,t) = (-15, 83, -157, 89, 24)
%C A131758 P(5,t) = (56, -424, 1266, -1724, 826, 120)
%C A131758 P(6,t) = (-185, 1887, -8038, 17642, -19593, 8287, 720)
%C A131758 P(7,t) = (204, -4976, 36226, -126944, 239576, -234688, 90602, 5040)
%C A131758 For the polynomial array:
%C A131758 1) The first column is A009940 = (-1)^n * n!*Lag(n,1) =(-1)^n* n!* Lag(n,
               -1,-1).
%C A131758 2) Row sums are n!.
%C A131758 3) Highest order coefficient is n!.
%C A131758 4) Alternating row sum is below.
%C A131758 Then, with Rf(n,t) = [ t/(1-t)^(n+1) ] * P(n,t)/n!, the polylogs are 
               given umbrally by
%C A131758 Li(-n,t)/n! = [ 1 + Rf(.,t) ]^n for n = 0,1,2,... so conversely
%C A131758 Rf(n,t) = {[ Li(-(.),t))/(.)! ]-1}^n.
%C A131758 Note umbrally [ Rf(.,t) ]^n = Rf(n,t) and
%C A131758 (1+Rf)^0 = 1^0 * [ Rf(.,t) ]^0 = Rf(0,t) = t/(1-t) = Li(0,t).
%C A131758 More generally, Newton interpolation holds and for Re(s)>= 0,
%C A131758 Li(-s,t)/(s)! = [ 1 + Rf(.,t) ]^s, when convergent in t.
%C A131758 Alternatively, the Rf's may be formed through differentiation of their 
               o.g.f., the Rgf(z,t) above, which may also be written as
%C A131758 Rgf(z,t) = sum(k=1,2,...) [ t^k ] * exp[ k * z/(z+1) ]/(z+1)
%C A131758 = sum(n=0,1,...) [ (-z)^n ] * sum(k=1,2,...)[ (t^k * Lag(n,k) ]
%C A131758 = sum(k=0,1,...) [ (-z)^k ] * Lag(k,Li(-(.),t))
%C A131758 = sum(k=0,1,...) [ z^k ] * {[ Li(-(.),t))/(.)! ]-1}^k
%C A131758 = exp[ Li(-(.),t)*z/(1+z) ]/(1+z),
%C A131758 and operationally as
%C A131758 Rgf(z,t) = {sum(k=0,1,...) (-z)^k * Lag(k,tD)} [ t/(1-t) ]
%C A131758 = {sum(k=0,1,...) (-z)^k * Lag(k,T(.,:tD:))} [ t/(1-t) ]
%C A131758 = {sum(k=0,1,...) (-z)^k * sum(j=0,...) Lag(k,j) (tD)^j /j!} [ x/(1-x) 
               ]
%C A131758 where D is w.r.t. x at 0
%C A131758 = {sum(k=0,1,...)(-z)^k*sum(j=0,...,k)(-1)^j*[ 1-Lag(k,.) ]^j*(:tD:)^j/ 
               j!} [ t/(1-t) ]
%C A131758 = {sum(k=0,1,...) (-z)^k * exp[ -[ 1-Lag(k,.) ]* :tD: ]} [ t/(1-t) ]
%C A131758 where (:tD:)^n = t^n * D^n, D is the derivative w.r.t. t unless otherwise 
               stated, Lag(n,x) is a Laguerre polynomial and T(n,t) is a Touchard 
               / Bell / exponential polynomial.
%C A131758 Hence [ t/(1-t)^(n+1) ] * P(n,t)/n! = Rf(n,t)
%C A131758 = {sum(k=0,...,n) (-1)^n-k)*[ C(m,k)/k! ]*(tD)^k} [ t/(1-t) ]
%C A131758 = {sum(k=0,...n) (-1)^(n-k)*[ C(m,k)/k! ]*sum(j=0,...,k)S2(k,j)*(:tD:)^j} 
               [ t/(1-t) ]
%C A131758 = {sum(k=0,1,...) (-1)^(n-k) * Lag(n,k) * (tD)^k/k!} [ x/(1-x) ] where 
               D is w.r.t. x at 0
%C A131758 = {sum(k=0,...,n) (-1)^(n-k)* [ 1-Lag(n,.) ]^k *(:tD:)^k/k!}[ t/(1-t) 
               ],
%C A131758 where S2(k,j) are the Stirling numbers of the second kind and C(m,k), 
               binomial coefficients.
%C A131758 The P(n,t) are related to the Laguerre polynomials through
%C A131758 P(n,t) = (-1)^n n! [ (1-t)^(n+1)} ] sum(k=0,1,...)[ (t^k*Lag(n,k+1) ] 
               = sum(m=0,...,n) a(n,m) * t^m
%C A131758 where a(n,m)= (-1)^n n! [ sum(k=0,...,m) (-1)^k * C(n+1,k) *Lag(n,m-k+1) 
               ] .
%C A131758 Conjecture for the polynomial array:
%C A131758 The greatest common divisor of the coefficients of each polynomial is 
               given by a(n)/n where the a(n)'s are A060872 or, equivalently, by 
               b(n) of A038548.
%C A131758 Some e.g.f.'s for the Rf's are
%C A131758 exp[ -Rf(.,t)*z ] = exp{[ 1-Li(-(.),t)/(.)! ]*z}
%C A131758 = sum(n=0,1,...) { (z^n/n!) * sum(k=1,2,...) [ t^k * Lag(n,k) ] }
%C A131758 = sum(k=1,2,...) { t^k * (e^z) * J_0[ 2*sqrt(k*z)}
%C A131758 = sum(n=0,1,...){(-1)^n*(z^n/n!)*(z^/j!)*Lag(n,-1)*sum(k=1,2,...)[ t^k*k^n*(k+1)^j 
               ]}
%C A131758 where J_0(x) is the zeroth Bessel function of the first kind.
%C A131758 The expressions (:tD:)^j}[ t/(1-t) ] and the Laguerre polynomials are 
               intimately connected to Lah numbers and rook polynomials.
%C A131758 Some interesting relations to physics, probability and number theory 
               are, for abs(t)<1 and abs(z)<1 at least,
%C A131758 BE(t,z) = sum(k=0,1,...) [ (-z)^k ] *[ 1 + Rf(.,t) ]^k
%C A131758 = Rgf(-z/(1+z),t)/(1+z) = t/{exp(z)-t}, a Bose-Einstein distribution,
%C A131758 FD(t,z) = sum(k=0,1,...) [ (-z)^k+1 ] *[ 1 + Rf(.,-t) ]^k
%C A131758 = -Rgf(-z/(1+z),-t)/(1+z) = t/{exp(z)+t}, a Fermi-Dirac distribution
%C A131758 and as t tends to 1 from below, z*BE(t,z) tends to the Bernoulli e.g.f., 
               which is related by the Mellin transform to(s-1)!*Zeta(s). Taking 
               Mellin transforms of BE and FD w.r.t. z gives the polylogarithm over 
               different domains.
%C A131758 Since BE(2,z) is essentially the e.g.f for the ordered Bell numbers, 
               it follows that umbrally
%C A131758 n! * Lag(n,OB(.)) = P(n,2) and
%C A131758 n! * Lag(n,P(.,2)) = OB(n)
%C A131758 where OB(n) are the signed ordered Bell/Fubini numbers A000670.
%C A131758 I.e., P(n,2) and the ordered Bell numbers form a reciprocal Laguerre 
               combinatorial transform pair,
%C A131758 or, equivalently, P(n,2)/n! and OB(n)/n! form a reciprocal finite difference 
               pair, so
%C A131758 P(n,2)/n! = (-1)^(n+1) * Rf(n,2) = -{1-[ Li(-(.),2))/(.)! ]}^n and
%C A131758 OB(n) = - Li(-n,2).
%C A131758 Note that n!*Lag(n,(.)!*Lag(.,x)) = x^n is a true identity for general 
               Laguerre polynomials Lag(n,x,a) with a = -1,0,1,..., so one could 
               look at analogous higher order reciprocal pairs containing OB(n).
%C A131758 In addition, a mixed-order iterated Laguerre transform gives
%C A131758 n!*Lag{n,(.)!*Lag[ .,P(.,2),0 ],-1}
%C A131758 = P(n,2) - n*P(n-1,2)
%C A131758 = n!*Lag[ n,OB(.),-1 ] = A084358(n), lists of sets of lists.
%C A131758 The relation to the Eulerian polynomials, E(n,t), given by A008292, is
%C A131758 E(n,t)/n! = [ 1-t+P(.,t)/(.)! ]^n
%C A131758 P(n,t)/n! = [ E(.,t)/(.)!-(1-t) ]^n, or equivalently
%C A131758 [ E(.,t)/(1-t) ]^n = n!*Lag[ n,-P(.,t)/(1-t) ]
%C A131758 [ -P(.,t)/(1-t) ]^n = n!*Lag[ n,E(.,t)/(1-t) ], a Laguerre transform 
               pair.
%C A131758 Then from known relations for the Eulerian polynomials, the alternating 
               row sum of the polynomial array is
%C A131758 P(n,-1) = (-2)^(n+1) * n! * Lag[ n,c(.)*Zeta(-(.)) ]
%C A131758 where c(n) = [ 2^(n+1) - 1 ] and Zeta is the Riemann zeta function. And 
               so
%C A131758 Zeta(-n) = n! * Lag[ n,-P(.,-1)/2 ] / [ 2 - 2^(n+2) ],
%C A131758 which also holds, with the summation limit of Lag extended to infinity, 
               for n = s, any complex number with Re(s)>0.
%C A131758 Then from standard formulas for the signed Euler numbers EN(n), the Bernoulli 
               numbers Ber(n), the Genocchi numbers GN(n), the Euler polynomials 
               EP(n,t), the Eulerian polynomials E(n,t), the Touchard / Bell polynomials 
               T(n,t) and the binomial C(x,y) =x!/[ (x-y)!*y! ]
%C A131758 2^(n+1) * (1-2^(n+1)) * (-1)^n * Zeta(-n)
%C A131758 = 2^(n+1) * (1-2^(n+1)) * Ber(n+1)/(n+1)
%C A131758 = [ -(1+EN(.)) ]^n
%C A131758 = 2^n * GN(n+1)/(n+1)
%C A131758 = 2^n * EP(n,0)
%C A131758 = (-1)^n * E(n,-1)
%C A131758 = (-2)^n * n! * Lag[ n,-P(.,-1)/2 ]
%C A131758 = (-2)^n * n! * C{T[ .,P(.,-1)/2 ] + n, n}
%C A131758 = an integer = Q(n)
%C A131758 These are related to the zag numbers A000182 by Zag(n) =abs[ Q(2*n-1) 
               ]. And, abs[ Q(2*n-1) ]/ 2^q(n) = Zag(n)/ 2^q(n) =A002425(n) with 
               q(n) = A101921 .
%C A131758 These may be generalized by letting n = s, a complex number, (or interpolating) 
               to obtain generalized Laguerre functions or confluent hypergeometric 
               functions of the first kind, M(a,b,x), or second kind, U(a,b,x), 
               whose arguments are P(.,-1)/2, such as,
%C A131758 E(s,-1)/[ 2^s*s! ] = -2*Li(-s,-1)/s! = (2-2^(s+2)) * Zeta(-s)/s!
%C A131758 = C{T[ .,P(.,-1)/2 ] + s, s} = Lag[ s,-P(.,-1)/2 ] =M[ -s,1,-P(.,-1)/
               2 ] or,
%C A131758 GN(s+1)/(s+1)! = EP(s,0)/s! = C{-T[ .,P(.,-1)/2 ]-1, n} = U[ -s,1,-P(.,
               -1)/2 ]/(s)!
%C A131758 And even more generally
%C A131758 E(s,t)/(1-t)^s = [ (1-t)/t ] Li(-s,t) = s!*Lag[ s,1,-P(.,t)/(1-t) ]
%C A131758 = s! * C{T[ .,P(.,t)/(1-t) ] + s, s} = s! * M[ -s,1,-P(.,t)/(1-t) ] .
%C A131758 The Laguerre polynomial expressions are fundamental as they can be interpolated 
               to form general M[ a,b,-P(.,t)/(1-t) ] or U[ a,b,-P(.,t)/(1-t) ] 
               which can then be related either directly or by binomial transforms 
               to many important Sheffer sequences, not to mention group theory 
               and Riemann surfaces.
%C A131758 Note for frequently occurring expressions above: The Laguerre polynomials 
               of order -1 and 0 are intimately connected to Lah numbers and rook 
               polynomials and (tD)^n [t/(1-t)] = T(n,:tD:) [t/(1-t)] generates 
               an eulerian polynomial in the numerator of a rational function. [From 
               Tom Copeland (tcjpn(AT)msn.com), Sep 09 2008]
%F A131758 a(n,m)= (-1)^n n! [ sum(k=0,...,m) (-1)^k * C(n+1,k) *Lag(n,m-k+1) ]
%Y A131758 Cf. A133289, A131202.
%Y A131758 Sequence in context: A091957 A056678 A092665 this_sequence A095909 A151872 
               A087420
%Y A131758 Adjacent sequences: A131755 A131756 A131757 this_sequence A131759 A131760 
               A131761
%K A131758 sign,tabl
%O A131758 0,6
%A A131758 Tom Copeland (tcjpn(AT)msn.com), Sep 17 2007, Sep 27 2007, Sep 30 2007, 
               Oct 01 2007, Oct 08 2007

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