0,2

a(n,m)= ^5P_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-(5+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(5*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+4)*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)^5).

Triangle (signed) = [ -5, -1, -6, -2, -7, -3, -8, -4, -9, ...] DELTA A000035; triangle (unsigned) = [5, 1, 6, 2, 7, 3, 8, 4, 9, ...] DELTA A000035; where DELTA is Deleham's operator defined in A084938.

{1}; {-5,1}; {30,-11,1}; {-210,107,-18,1}; ... s(2,x)= 30-11*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Unsigned column sequences are: A001720-A001724. Row sums (signed triangle): A001715(n+3)*(-1)^n. Row sums (unsigned triangle): A001725(n+5).

Cf. A000035 A084938.

Sequence in context: A140713 A125906 A135892 this_sequence A062140 A049353 A027759

Adjacent sequences: A049457 A049458 A049459 this_sequence A049461 A049462 A049463

sign,easy,tabl

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