%I A008517
%S A008517 1,1,2,1,8,6,1,22,58,24,1,52,328,444,120,1,114,1452,4400,3708,720,1,
%T A008517 240,5610,32120,58140,33984,5040,1,494,19950,195800,644020,785304,
%U A008517 341136,40320,1,1004,67260,1062500,5765500,12440064,11026296,3733920
%N A008517 Second-order Eulerian triangle T(n,k), 1<=k<=n.
%C A008517 When seen as coefficients of polynomials with descending exponents, evaluations 
               are in A000311 (x=2) and A001662 (x=-1).
%C A008517 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|.
%C A008517 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).
%C A008517 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.
%C A008517 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.
%C A008517 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).
%C A008517 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
%C A008517 From the recurrence relation, the generating function F(x,y) := 1 + Sum[T(n,
               k)x^n/n!*y^k,{n>=1,1<=k<=n}] satisfies the partial differential equation 
               F = (1/y-2x)F_x + (y-1)F_y, with (non-elementary) solution F(x,y) 
               = (1-y)/(1-Phi(w)) where w = y*exp(x(y-1)^2-y) and Phi(x) is defined 
               by Phi(x) = x*exp(Phi(x)). By Lagrange inversion (see Wilf's book 
               "generatingfunctionology", page 168, Example 1), Phi(x) = Sum[n^(n-1)*x^n/
               n!,{n>=1}]. Thus Phi(x) can alternatively be described as the egf 
               for rooted labeled trees on n vertices A000169. - David Callan (callan(AT)stat.wisc.edu), 
               Jul 25 2008
%C A008517 A method for solving PDEs such as the one above for F(x,y) is described 
               in the Klazar reference (see pages 207-208). In his case, the auxiliary 
               ODE dy/dx = b(x,y)/a(x,y) is exact; in this case it is not exact 
               but has an integrating factor depending on y alone, namely y-1. The 
               egf for the row sums A001147 is 1/Sqrt[1-2x] and the demonstration 
               that F(x,1) = 1/Sqrt[1-2x] is interesting: two applications of l'Hopital's 
               rule to lim_{y->1}F(x,y) yield F(x,1) = 1/(1-2x)*1/F(x,1). So l'Hopital's 
               rule doesn't directly yield F(x,1) but rather an equation to be solved 
               for F(x,1)! - David Callan (callan(AT)stat.wisc.edu), Jul 25 2008
%C A008517 Comment from Tom Copeland (tcjpn(AT)msn.com), Oct 12 2008: (Start)
%C A008517 Let P(0,t)= 0, P(1,t)= 1, P(2,t)= t, P(3,t)= t + 2 t^2, P(4,t)= t + 8 
               t + 6 t^2, ... row polynomials of top array, then
%C A008517 exp[x*P(.,t)] = { u + Tree[t*exp(u)] } / (1-t) = WD[x*(1-t), t/(1-t)] 
               / (1-t)
%C A008517 where u = x*(1-t)^2 - t, Tree(x) is the e.g.f. of A000169 and WD(x,t) 
               is the e.g.f. for A134991, relating the Ward and 2-Eulerian polynomials 
               by a simple transformation.
%C A008517 Note also apparently P(4,t) / (1-t)^3 = Ward Poly(4, t/(1-t)) = essentially 
               an e.g.f. for A093500.
%C A008517 The compositional inverse of f(x,t) = exp[P(.,t)*x] about x=0 is
%C A008517 g(x,t) = { x - [t/(1-t)^2][exp[x*(1-t)]-x*(1-t)-1] }
%C A008517 = x - t x^2/2! - t(1-t) x^3/3! - t(1-t)^2 x^4/4! - t(1-t)^3 x^5/5! - 
               ... .
%C A008517 Can apply A134685 to these coefficients to generate f(x,t). (End)
%C A008517 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%C A008517 Triangle A163936 is similar to the one given above except for an extra 
               right hand column [1, 0, 0, 0, ... ] and that its row order is reversed.
%C A008517 (End)
%D A008517 I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, 
               A 24 (1978), 24-33.
%D A008517 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 256.
%D A008517 O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk 
               Matematisk Tidskrift, 7 (1959), 5-19.
%D A008517 Martin Klazar, Twelve countings with rooted plane trees, Europ. J. Combinatorics, 
               Vol. 18, 1997, 195-210.
%D A008517 L. Carlitz, The coefficients in an asymptotic expansion. Proc. Amer. 
               Math. Soc. 16 (1965) 248-252. [From David Callan (callan(AT)stat.wisc.edu), 
               Aug 27 2009]
%D A008517 J. Ginsburg, Note on Stirling's Numbers. Amer. Math. Monthly 35 (1928), 
               no. 2, 77-80. [From David Callan (callan(AT)stat.wisc.edu), Aug 27 
               2009]
%H A008517 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Second-OrderEulerianTriangle.html">Second-Order Eulerian Triangle</
               a>
%F A008517 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%F A008517 a(n,m) = sum((-1)^(n+k)*binomial(2*n+1,k)*stirling1(2*n-m-k+1,n-m-k+1), 
               k=0..n-m)
%F A008517 (End)
%e A008517 1; 1,2; 1,8,6; 1,22,58,24; 1,52,328,444,120; ...
%p A008517 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%p A008517 with(combinat, stirling1): nmax:=9; for n from 1 to nmax do for m from 
               1 to n do a(n,m):=sum((-1)^(n+k)*binomial(2*n+1,k)*stirling1(2*n-m-k+1,
               n-m-k+1),k=0..n-m) od: od: T:=1: for n from 1 to nmax do for m from 
               1 to n do a(T):=a(n,m): T:=T+1: od: od: seq(a(n),n=1..T-1); for n 
               from 1 to nmax do seq(a(n,m),m=1..n) od;
%p A008517 (End)
%o A008517 (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))
%Y A008517 Columns include A005803, A004301, A006260. Right-hand columns include 
               A000142, A002538, A002539. Row sums are A001147.
%Y A008517 Sequence in context: A136230 A004732 A011244 this_sequence A142336 A114193 
               A039683
%Y A008517 Adjacent sequences: A008514 A008515 A008516 this_sequence A008518 A008519 
               A008520
%K A008517 nonn,tabl,nice,easy
%O A008517 1,3
%A A008517 N. J. A. Sloane (njas(AT)research.att.com).
%E A008517 More terms from Michael Somos, Oct 13, 2002