1,3

When seen as coefficients of polynomials with descending exponents, evaluations are in A000311 (x=2) and A001662 (x=-1).

The row reversed triangle is A112007. There one can find comments on the o.g.f.s for the diagonals of the unsigned Stirling1 triangle |A008275|.

Stirling2(n,n-k) = sum(T(k,m+1)*binomial(n+k-1+m,2*k),m=0..k-1) k>=1. See the Graham et al. reference p. 257 eq. (6.43).

This triangle is the coefficient triangle of numerator polynomials appearing in the o.g.f. for the k-th diagonal of the Stirling2 triangle A048993.

The o.g.f. for column k satisfies the recurrence G(k,x)= x*(2*x*diff(G(k-1,x),x) + (2-k)*G(k-1,x))/(1-k*x),k>=2, with G(1,x)=1/(1-x). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 14 2005.

T(n,k) = 0 if n<k, T(1,1)=1, T(n,-1):=0, T(n,k)=k*T(n-1,k)+(2*n-k)*T(n-1,k-1).

This triangle is in some sense generated by the differential equation y' = 1-2/(1+x+y). (This is the differential equation satisfied by the function defined implicitly as x+y=exp(x-y). ) If we take y = a(0) + a(1)x + a(2)x^2 + a(3)x^3 +... and assume a(0)=c then all the a's may be calculated formaly in terms of c, and we have a(1)= (c-1)/(c+1) and for n>1 a(n) = 2^n/n! (1+c)^(1-2n)( T(n,1)c - T(n,2)c^2 + T(n,3)c^3...+ (-1)^(n-1) T(n,n)c^n ) - Moshe Newman (mshnoiman(AT)hotmail.com), Aug 08 2007

I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 256.

O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.

Eric Weisstein's World of Mathematics, Second-Order Eulerian Triangle

1; 1,2; 1,8,6; 1,22,58,24; 1,52,328,444,120; ...

(PARI) T(n, k)=if(n<1, 0, z=1+O(x); for(k=1, n, z=1+intformal(z^2*(z+y-1))); n!*polcoeff(polcoeff(z, n), k))

Columns include A005803, A004301, A006260. Right-hand columns include A000142, A002538, A002539. Row sums are A001147.

Adjacent sequences: A008514 A008515 A008516 this_sequence A008518 A008519 A008520

Sequence in context: A136230 A004732 A011244 this_sequence A114193 A039683 A108084

nonn,tabl,nice,easy

njas

More terms from Michael Somos, Oct 13, 2002