0,2

a(n,m)= ^4P_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-(4+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(4*t),exp(t)-1).

See A143493 for the unsigned version of this array and A143496 for the inverse. [From Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008]

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+3)*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)^4).

Triangle (signed) = [ -4, -1, -5, -2, -6, -3, -7, -4, -8, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [4, 1, 5, 2, 6, 3, 7, 4, 8, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deleham's operator defined in A084938 (unsigned version in A143493).

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,4), for n=1,2,...;i=0...n. [From Milan R. Janjic (agnus(AT)blic.net), Dec 21 2008]

{1}; {-4,1}; {20,-9,1}; {-120,74,-15,1}; ...

Unsigned column sequences are: A001715-A001719. Cf. A008275 (Stirling1 triangle, A049458, A049460. Row sums (signed triangle): A001710(n+2)*(-1)^n. Row sums (unsigned triangle): A001720(n+4).

Cf. A000035 A084938.

A143493, A143496. [From Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008]

Sequence in context: A167432 A078939 A135891 this_sequence A143493 A062137 A143497

Adjacent sequences: A049456 A049457 A049458 this_sequence A049460 A049461 A049462

sign,easy,tabl

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

Corrected second formula. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 09 2008