%I A049444
%S A049444 1,2,1,6,5,1,24,26,9,1,120,154,71,14,1,720,1044,580,155,20,1,5040,
%T A049444 8028,5104,1665,295,27,1,40320,69264,48860,18424,4025,511,35,1,362880,
%U A049444 663696,509004,214676,54649,8624,826,44,1,3628800,6999840,5753736
%V A049444 1,-2,1,6,-5,1,-24,26,-9,1,120,-154,71,-14,1,-720,1044,-580,155,-20,1,
               5040,
%W A049444 -8028,5104,-1665,295,-27,1,-40320,69264,-48860,18424,-4025,511,-35,1,
               362880,
%X A049444 -663696,509004,-214676,54649,-8624,826,-44,1,-3628800,6999840,-5753736
%N A049444 Generalized Stirling number triangle of first kind.
%C A049444 a(n,m)= ^2P_n^m in the notation of the given reference with a(0,0) := 
               1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which 
               are s(n,x)= product(x-(2+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy 
               s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 
               polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
%C A049444 In the umbral calculus (see the S. Roman reference given in A048854) 
               the s(n,x) polynomials are called Sheffer polynomials for (exp(2*t),
               exp(t)-1).
%C A049444 See A143491 for the unsigned version of this array and A143494 for the 
               inverse. [From Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008]
%D A049444 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres 
               relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. 
               Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%D A049444 Y. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, 
               American Math. Soc. Colloquium Publications Vol. 47, 1999
%F A049444 a(n, m)= a(n-1, m-1) -(n+1)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; 
               a(n, -1) := 0, a(0, 0)=1.
%F A049444 E.g.f. for m-th column of signed triangle: ((ln(1+x))^m)/(m!*(1+x)^2).
%F A049444 Triangle (signed) = [ -2, -1, -3, -2, -4, -3, -5, -4, -6, -5, ...] DELTA 
               [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [2, 1, 3, 2, 
               4, 3, 5, 4, 6, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...], 
               where DELTA is Deleham's operator defined in A084938 (unsigned version 
               in A143491).
%F A049444 E.g.f.: (1+y)^(x-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 17 
               2004
%F A049444 With P(n,t) = sum(j=0,...,n-2) a(n-2,j) * t^j and P(1,t) = -1 and P(0,
               t) = 1, then G(x,t) = -1 + exp[P(.,t)*x] = [(1+x)^t - 1 - t^2 * x] 
               / [t(t-1)], whose compositional inverse in x about 0 is given in 
               A074060. G(x,0) = -ln(1+x) and G(x,1) = (1+x) ln(1+x) - 2x . G(x,
               q^2) occurs in formulae on pages 194-196 of the Manin reference. 
               - Tom Copeland (tcjpn(AT)msn.com), Feb 17 2008
%F A049444 If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,
               j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,2), for n=1,2,...;i=0...n 
               [From Milan R. Janjic (agnus(AT)blic.net), Dec 21 2008]
%e A049444 {1}; {-2,1}; {6,-5,1}; {-24,26,-9,1}; ...
%e A049444 1; -1,1; 0,-1,1; 0,-1,0,1; 0,-2,-1,2,1; 0,-6,-5,5,5,1; 0,-24,-26,15,25,
               9,1; ...
%Y A049444 Unsigned column sequences are A000142(n+1), A001705-A001709. Row sums 
               (signed triangle): n!*(-1)^n, row sums (unsigned triangle): A001710(n-2). 
               Cf. A008275 (Stirling1 triangle).
%Y A049444 Cf. A000035 A084938, A094645, A094646.
%Y A049444 Cf. A143491, A143494. [From Peter Bala (pbala(AT)toucansurf.com), Aug 
               25 2008]
%Y A049444 Sequence in context: A133367 A121576 A121575 this_sequence A136124 A143491 
               A070918
%Y A049444 Adjacent sequences: A049441 A049442 A049443 this_sequence A049445 A049446 
               A049447
%K A049444 sign,easy,tabl,nice
%O A049444 0,2
%A A049444 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
%E A049444 Corrected second formula. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 09 2008