0,2

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).

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.

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.

E.g.f. for m-th column of signed triangle: ((ln(1+x))^m)/(m!*(1+x)^6).

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.

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]

{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).

Unsigned m=0 column sequence is: A001725. Row sums (signed triangle): A001720(n+4)*(-1)^n. Row sums (unsigned triangle): A001730(n+6).

Cf. A000035 A084938.

Sequence in context: A145357 A035529 A135893 this_sequence A062138 A143498 A144356

Adjacent sequences: A051335 A051336 A051337 this_sequence A051339 A051340 A051341

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)