%I A051338
%S A051338 1,6,1,42,13,1,336,146,21,1,3024,1650,335,30,1,30240,19524,
%T A051338 5000,635,40,1,332640,245004,74524,11985,1075,51,1,3991680,
%U A051338 3272688,1139292,218344,24885,1687,63,1,51891840,46536624,18083484,3977764,
               541849,46816,2506,76,1
%V A051338 1,-6,1,42,-13,1,-336,146,-21,1,3024,-1650,335,-30,1,-30240,
%W A051338 19524,-5000,635,-40,1,332640,-245004,74524,-11985,1075,-51,1,
%X A051338 -3991680,3272688,-1139292,218344,-24885,1687,-63,1,51891840,-46536624,
               18083484,-3977764,541849,-46816,2506,-76,1
%N A051338 Generalized Stirling number triangle of first kind.
%C A051338 a(n,m)= ^6P_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-(6+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. 
               In the umbral calculus (see the S. Roman reference given in A048854) 
               the s(n,x) polynomials are called Sheffer for (exp(6*t),exp(t)-1).
%D A051338 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.
%F A051338 a(n, m)= a(n-1, m-1) - (n+5)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; 
               a(n, -1) := 0, a(0, 0)=1.
%F A051338 E.g.f. for m-th column of signed triangle: ((ln(1+x))^m)/(m!*(1+x)^6).
%F A051338 Triangle (signed) = [ -6, -1, -7, -2, -8, -3, -9, -4, -10, ...] DELTA 
               A000035; triangle (unsigned) = [6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, 
               ...] DELTA A000035; where DELTA is Deleham's operator defined in 
               A084938.
%F A051338 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,6), for n=1,2,...;i=0...n. 
               [From Milan R. Janjic (agnus(AT)blic.net), Dec 21 2008]
%e A051338 {1}; {-6,1}; {42,-13,1}; {-336,146,-21,1}; ... s(2,x)= 42-13*x+x^2; S1(2,
               x)= -x+x^2 (Stirling1).
%Y A051338 Unsigned m=0 column sequence is: A001725. Row sums (signed triangle): 
               A001720(n+4)*(-1)^n. Row sums (unsigned triangle): A001730(n+6).
%Y A051338 Cf. A000035 A084938.
%Y A051338 Sequence in context: A145357 A035529 A135893 this_sequence A062138 A143498 
               A144356
%Y A051338 Adjacent sequences: A051335 A051336 A051337 this_sequence A051339 A051340 
               A051341
%K A051338 sign,easy,tabl
%O A051338 0,2
%A A051338 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)