%I A111596
%S A111596 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,
%T A111596 0,5040,15120,12600,4200,630,42,1,0,40320,141120,141120,58800,11760,1176,
               56,1,0,
%U A111596 362880,1451520,1693440,846720,211680,28224,2016,72,1,0,3628800
%V A111596 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,
%W A111596 0,5040,-15120,12600,-4200,630,-42,1,0,-40320,141120,-141120,58800,-11760,
               1176,-56,1,0,
%X A111596 362880,-1451520,1693440,-846720,211680,-28224,2016,-72,1,0,-3628800
%N A111596 Associated triangle to Sheffer triangle A111595.
%C A111596 Without row n=0 and column m=0 this is, up to signs, the Lah triangle 
               A008297.
%C A111596 The unsigned column sequences are (with leading zeros): A000142, A001286, 
               A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.
%C A111596 The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the 
               row polynomials s(n,x) of A111595 satisfy the exponential (or binomial) 
               convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),
               k=0..n), n>=0.
%C A111596 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 (pbarry(AT)wit.ie), Apr 12 2007
%C A111596 For the unsigned subtriangle without column nr. m=0 and row nr. n=0 see 
               A105278.
%C A111596 Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.
%C A111596 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 formulae. - Tom Copeland (tcjpn(AT)msn.com), Nov 17 
               2007, Sep 09 2008
%C A111596 An infinitesimal matrix generator for unsigned A111596 is given by A132792. 
               - Tom Copeland (tcjpn(AT)msn.com), Nov 22 2007
%C A111596 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 (tcjpn(AT)msn.com), Nov 22 2007
%C A111596 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. An e.g.f. for the row polynomials of A is exp[ -a(.)*x^2*D_x] 
               exp(x*t) = exp(x*t) exp[(.)!*Lag(.,-x*t,-1)*a(.)*(-x)], umbrally, 
               where [(.)! Lag(.,x,-1)]^n = n! Lag(n,x,-1) is a normalized Laguerre 
               polynomial of order -1. (Caution: sometimes the factor n! is included 
               in the definition of the Laguerre polynomial (Rota convention), as 
               in an earlier comment in this entry.) [From Tom Copeland (tcjpn(AT)msn.com), 
               Aug 27 2008]
%H A111596 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A111596 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A111596.text">
               First 10 rows.</a>
%F A111596 E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.
%F A111596 E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).
%F A111596 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 A111596 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 A111596 |a(n,m)|=sum(|S1(n,k)|*S2(k,m),k=m..n), n>=0. S2(n,m):=A048993. S1(n,
               m):=A048994. W. Lang, May 04, 2007.
%e A111596 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 A111596 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 A111596 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) +
%e A111596 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)
%e A111596 + 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.
%Y A111596 Row sums: A111884. Unsigned row sums: A000262.
%Y A111596 Sequence in context: A047922 A021830 A111184 this_sequence A129062 A163936 
               A117651
%Y A111596 Adjacent sequences: A111593 A111594 A111595 this_sequence A111597 A111598 
               A111599
%K A111596 sign,easy,tabl
%O A111596 0,5
%A A111596 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Aug 23 2005