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 A111596 The matrix inverse of the unsigned Lah numbers A271703. 43
 1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1, 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also the associated Sheffer triangle to Sheffer triangle A111595. 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 Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297. The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10. 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. 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 For the unsigned subtriangle without column number m=0 and row number n=0, see A105278. Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices. 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 An infinitesimal matrix generator for unsigned A111596 is given by A132792. - Tom Copeland, Nov 22 2007 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 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 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 LINKS G. C. Greubel, Rows n=0..100 of triangle, flattened Wolfdieter Lang, The first 11 rows of the triangle. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4. Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6. Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 20. Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011)  11.6.7. Tom Copeland, A Class of Differential Operators and the Stirling Numbers; Generators, Inversion, and Matrix, Binomial, and Integral Transforms; Lagrange a la Lah A. Hennessy and P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2. M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3. Mathoverflow, Pochhammer symbol of a differential, and hypergeometric polynomials, a question posed by Emilio Pisanty and answered by Tom Copeland, 2012. J. Taylor, Counting words with Laguerre polynomials, DMTCS Proc., Vol. AS, 2013, p. 1131-1142. [Tom Copeland, Jan 08 2016] [Broken link] J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96. [Tom Copeland, Dec 20 2018] Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021. FORMULA E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0. E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)). 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. 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=0. S2(n,m):=A048993. S1(n,m):=A048994. - Wolfdieter Lang, May 04 2007 From Tom Copeland, Nov 21 2011: (Start) For this Lah triangle, the n-th row polynomial is given umbrally by (-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. 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) The n-th row polynomial is (-1/x)^n e^x (x^2*D_x)^n e^(-x). - Tom Copeland, Oct 29 2012 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 |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 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 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 EXAMPLE 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, 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 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. From Wolfdieter Lang, Apr 28 2014: (Start) The triangle a(n,m) begins: n\m 0 1 2 3 4 5 6 7 0: 1 1: 0 1 2: 0 -2 1 3: 0 6 -6 1 4: 0 -24 36 -12 1 5: 0 120 -240 120 -20 1 6: 0 -720 1800 -1200 300 -30 1 7: 0 5040 -15120 12600 -4200 630 -42 1 ... For more rows see the link. (End) MAPLE # The function BellMatrix is defined in A264428. BellMatrix(n -> `if`(n::odd, -(n+1)!, (n+1)!), 9); # Peter Luschny, Jan 27 2016 MATHEMATICA 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[ n_, k_] := (-1)^n n! Coefficient[ LaguerreL[ n, -1, x], x, k]; (* Michael Somos, Dec 15 2014 *) rows = 9; t = Table[(-1)^(n+1) n!, {n, 1, rows}]; T[n_, k_] := BellY[n, k, t]; Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *) PROG (Sage) lah_number = lambda n, k: factorial(n-k)*binomial(n, n-k)*binomial(n-1, n-k) A111596_row = lambda n: [(-1)^(n-k)*lah_number(n, k) for k in (0..n)] for n in range(10): print(A111596_row(n)) # Peter Luschny, Oct 05 2014 (Sage) # uses[inverse_bell_transform from A264429] def A111596_matrix(dim): fact = [factorial(n) for n in (1..dim)] return inverse_bell_transform(dim, fact) A111596_matrix(10) # Peter Luschny, Dec 20 2015 (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 */ CROSSREFS Row sums: A111884. Unsigned row sums: A000262. A002868 gives maximal element (in magnitude) in each row. Cf. A130561 for a natural refinement. Cf. A264428, A264429, A271703 (unsigned). Cf. A008297, A089231, A105278 (variants). Cf. A002378, A008277, A132792, A133314, A132792, A001263. Sequence in context: A247686 A352369 A111184 * A271703 A276922 A129062 Adjacent sequences: A111593 A111594 A111595 * A111597 A111598 A111599 KEYWORD sign,easy,tabl AUTHOR Wolfdieter Lang, Aug 23 2005 EXTENSIONS New name using a comment from Wolfdieter Lang by Peter Luschny, May 10 2021 STATUS approved

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