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The matrix inverse of the unsigned Lah numbers A271703.
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%I #187 Jan 18 2022 14:11:33

%S 1,0,1,0,-2,1,0,6,-6,1,0,-24,36,-12,1,0,120,-240,120,-20,1,0,-720,

%T 1800,-1200,300,-30,1,0,5040,-15120,12600,-4200,630,-42,1,0,-40320,

%U 141120,-141120,58800,-11760,1176,-56,1,0,362880,-1451520,1693440,-846720,211680,-28224,2016,-72,1

%N The matrix inverse of the unsigned Lah numbers A271703.

%C Also the associated Sheffer triangle to Sheffer triangle A111595.

%C Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-1,x), which equals (-1)^n * Lag(n,x,-1) below. Lag(n,Lag(.,x,-1),-1) = x^n evaluated umbrally, i.e., with (Lag(.,x,-1))^k = Lag(k,x,-1). - _Tom Copeland_, Apr 26 2014

%C Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297.

%C The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.

%C The row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m, together with the row polynomials s(n,x) of A111595 satisfy the exponential (or binomial) convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), n>=0.

%C Exponential Riordan array [1,x/(1+x)]. Inverse of the exponential Riordan array [1,x/(1-x)], which is the unsigned version of A111596. - _Paul Barry_, Apr 12 2007

%C For the unsigned subtriangle without column number m=0 and row number n=0, see A105278.

%C Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.

%C The unsigned row polynomials are Lag(n,-x,-1), the associated Laguerre polynomials of order -1 with negated argument. See Gradshteyn and Ryzhik, Abramowitz and Stegun and Rota (Finite Operator Calculus) for extensive formulas. - _Tom Copeland_, Nov 17 2007, Sep 09 2008

%C An infinitesimal matrix generator for unsigned A111596 is given by A132792. - _Tom Copeland_, Nov 22 2007

%C From the formalism of A132792 and A133314 for n > k, unsigned A111596(n,k) = a(k) * a(k+1)...a(n-1) / (n-k)! = a generalized factorial, where a(n) = A002378(n) = n-th term of first subdiagonal of unsigned A111596. Hence Deutsch's remark in A002378 provides an interpretation of A111596(n,k) in terms of combinations of certain circular binary words. - _Tom Copeland_, Nov 22 2007

%C Given T(n,k)= A111596(n,k) and matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - _Tom Copeland_, Aug 27 2008

%C Operationally, the unsigned row polynomials may be expressed as p_n(:xD:) = x*:Dx:^n*x^{-1}=x*D^nx^n*x^{-1}= n!*binomial(xD+n-1,n) = (-1)^n n! binomial(-xD,n) = n!L(n,-1,-:xD:), where, by definition, :AB:^n = A^nB^n for any two operators A and B, D = d/dx, and L(n,-1,x) is the Laguerre polynomial of order -1. A similarity transformation of the operators :Dx:^n generates the higher order Laguerre polynomials, which can also be expressed in terms of rising or falling factorials or Kummer's confluent hypergeometric functions (cf. the Mathoverflow post). - _Tom Copeland_, Sep 21 2019

%H G. C. Greubel, <a href="/A111596/b111596.txt">Rows n=0..100 of triangle, flattened</a>

%H Wolfdieter Lang, <a href="/A111596/a111596.pdf">The first 11 rows of the triangle.</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry3/barry100r.html">The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms</a>, J. Int. Seq. 13 (2010) # 10.8.4, example 4.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry4/barry122.html">Exponential Riordan Arrays and Permutation Enumeration</a>, J. Int. Seq. 13 (2010) # 10.9.1, example 6.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, example 20.

%H Paul Barry, <a href="http://arxiv.org/abs/1105.3044">Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays</a>, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry5/barry112.html">J. Int. Seq. 14 (2011) 11.6.7</a>.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/">A Class of Differential Operators and the Stirling Numbers</a>; <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>; <a href="https://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>

%H A. Hennessy and P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry6/barry161.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials</a>, J. Int. Seq. 14 (2011) # 11.8.2.

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3.

%H Mathoverflow, <a href="https://mathoverflow.net/questions/107159/pochhammer-symbol-of-a-differential-and-hypergeometric-polynomials/107191#107191">Pochhammer symbol of a differential, and hypergeometric polynomials</a>, a question posed by Emilio Pisanty and answered by Tom Copeland, 2012.

%H J. Taylor, <a href="https://hal.inria.fr/hal-01229684">Counting words with Laguerre polynomials</a>, DMTCS Proc., Vol. AS, 2013, p. 1131-1142. [_Tom Copeland_, Jan 08 2016] [Broken link]

%H J. Taylor, <a href="https://digital.lib.washington.edu/researchworks/handle/1773/36757">Formal group laws and hypergraph colorings</a>, doctoral thesis, Univ. of Wash., 2016, p. 96. [_Tom Copeland_, Dec 20 2018]

%H Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.

%F E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.

%F E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).

%F a(n, m) = ((-1)^(n-m))*|A008297(n, m)| = ((-1)^(n-m))*(n!/m!)*binomial(n-1, m-1), n>=m>=1; a(0, 0)=1; else 0.

%F a(n, m) = -(n-1+m)*a(n-1, m) + a(n-1, m-1), n>=m>=0, a(n, -1):=0, a(0, 0)=1; a(n, m)=0 if n<m.

%F |a(n,m)| = Sum_{k=m..n} |S1(n,k)|*S2(k,m), n>=0. S2(n,m):=A048993. S1(n,m):=A048994. - _Wolfdieter Lang_, May 04 2007

%F From _Tom Copeland_, Nov 21 2011: (Start)

%F For this Lah triangle, the n-th row polynomial is given umbrally by

%F (-1)^n n! binomial(-Bell.(-x),n), where Bell_n(-x)= exp(x)(xd/dx)^n exp(-x), the n-th Bell / Touchard / exponential polynomial with neg. arg., (cf. A008277). E.g., 2! binomial(-Bell.(-x),2) = -Bell.(-x)*(-Bell.(-x)-1) = Bell_2(-x)+Bell_1(-x) = -2x+x^2.

%F A Dobinski relation is (-1)^n n! binomial(-Bell.(-x),n)= (-1)^n n! e^x Sum_{j>=0} (-1)^j binomial(-j,n)x^j/j!= n! e^x Sum_{j>=0} (-1)^j binomial(j-1+n,n)x^j/j!. See the Copeland link for the relation to inverse Mellin transform. (End)

%F The n-th row polynomial is (-1/x)^n e^x (x^2*D_x)^n e^(-x). - _Tom Copeland_, Oct 29 2012

%F Let f(.,x)^n = f(n,x) = x!/(x-n)!, the falling factorial,and r(.,x)^n = r(n,x) = (x-1+n)!/(x-1)!, the rising factorial, then the Lah polynomials, Lah(n,t)= n!*Sum{k=1..n} binomial(n-1,k-1)(-t)^k/k! (extra sign factor on odd rows), give the transform Lah(n,-f(.,x))= r(n,x), and Lah(n,r(.,x))= (-1)^n * f(n,x). - _Tom Copeland_, Oct 04 2014

%F |T(n,k)| = Sum_{j=0..2*(n-k)} A254881(n-k,j)*k^j/(n-k)!. Note that A254883 is constructed analogously from A254882. - _Peter Luschny_, Feb 10 2015

%F The T(n,k) are the inverse Bell transform of [1!,2!,3!,...] and |T(n,k)| are the Bell transform of [1!,2!,3!,...]. See A264428 for the definition of the Bell transform and A264429 for the definition of the inverse Bell transform. - _Peter Luschny_, Dec 20 2015

%F Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates a shifted, signed Narayana matrix A001263. - _Tom Copeland_, Sep 23 2020

%e Binomial convolution of row polynomials: p(3,x) = 6*x-6*x^2+x^3; p(2,x) = -2*x+x^2, p(1,x) = x, p(0,x) = 1,

%e together with those from A111595: s(3,x) = 9*x-6*x^2+x^3; s(2,x) = 1-2*x+x^2, s(1,x) = x, s(0,x) = 1; therefore

%e 9*(x+y)-6*(x+y)^2+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = (6*y-6*y^2+y^3) + 3*x*(-2*y+y^2) + 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.

%e From _Wolfdieter Lang_, Apr 28 2014: (Start)

%e The triangle a(n,m) begins:

%e n\m 0 1 2 3 4 5 6 7

%e 0: 1

%e 1: 0 1

%e 2: 0 -2 1

%e 3: 0 6 -6 1

%e 4: 0 -24 36 -12 1

%e 5: 0 120 -240 120 -20 1

%e 6: 0 -720 1800 -1200 300 -30 1

%e 7: 0 5040 -15120 12600 -4200 630 -42 1

%e ...

%e For more rows see the link.

%e (End)

%p # The function BellMatrix is defined in A264428.

%p BellMatrix(n -> `if`(n::odd, -(n+1)!, (n+1)!), 9); # _Peter Luschny_, Jan 27 2016

%t a[0, 0] = 1; a[n_, m_] := ((-1)^(n-m))*(n!/m!)*Binomial[n-1, m-1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 05 2013 *)

%t T[ n_, k_] := (-1)^n n! Coefficient[ LaguerreL[ n, -1, x], x, k]; (* _Michael Somos_, Dec 15 2014 *)

%t rows = 9;

%t t = Table[(-1)^(n+1) n!, {n, 1, rows}];

%t T[n_, k_] := BellY[n, k, t];

%t Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)

%o (Sage)

%o lah_number = lambda n, k: factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)

%o A111596_row = lambda n: [(-1)^(n-k)*lah_number(n, k) for k in (0..n)]

%o for n in range(10): print(A111596_row(n)) # _Peter Luschny_, Oct 05 2014

%o (Sage) # uses[inverse_bell_transform from A264429]

%o def A111596_matrix(dim):

%o fact = [factorial(n) for n in (1..dim)]

%o return inverse_bell_transform(dim, fact)

%o A111596_matrix(10) # _Peter Luschny_, Dec 20 2015

%o (PARI) {T(n, k) = if( n<1 || k<1, n==0 && k==0, (-1)^n * n! * polcoeff( sum(k=1, n, binomial( n-1, k-1) * (-x)^k / k!), k))}; /* _Michael Somos_, Dec 15 2014 */

%Y Row sums: A111884. Unsigned row sums: A000262.

%Y A002868 gives maximal element (in magnitude) in each row.

%Y Cf. A130561 for a natural refinement.

%Y Cf. A264428, A264429, A271703 (unsigned).

%Y Cf. A008297, A089231, A105278 (variants).

%Y Cf. A002378, A008277, A132792, A133314, A132792, A001263.

%K sign,easy,tabl

%O 0,5

%A _Wolfdieter Lang_, Aug 23 2005

%E New name using a comment from _Wolfdieter Lang_ by _Peter Luschny_, May 10 2021