1,5

O.g.f. for the k-th column sequence is 1/(1-x) if k=1 and A(k,x):=((x^k)/(1-x)^(2*k+1))*sum(A008517(k,m+1)*x^m,m=0..k-1) if k>=2. A008517 is the second-order Eulerian triangle. Cf. p. 257, eq. (6.43) of the R. L. Graham et al. book. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 14 2005.

E.g.f. for the k-th column sequence with offset n=0 is E(k,x):= exp(x)*sum(A112493(k-1,m)*(x^(k-1+m))/(k-1+m)!,m=0..k-1) if k>=1. W. Lang, Oct 14 2005.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994.

T. D. Noe, Rows n=0..100 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

a(n, k)=0 if n<k, a(n, 0)=0, a(1, 1)=1, a(n, k)= (n-k+1)*a(n-1, k-1) + a(n-1, k) else.

The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).

See A008277 and A048993, which are the main entries for this triangle of numbers.

Cf. A094262, A008277, A008276, A003422, A000166, A000110, A000204, A000045, A000108.

Sequence in context: A146769 A143362 A133713 this_sequence A056858 A137251 A158359

Adjacent sequences: A008275 A008276 A008277 this_sequence A008279 A008280 A008281

nonn,tabl,nice

N. J. A. Sloane (njas(AT)research.att.com).