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

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

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

Comment from Tom Copeland (tcjpn(AT)msn.com), Oct 12 2008: (Start)

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

exp[x*P(.,t)] = { u + Tree[t*exp(u)] } / (1-t) = WD[x*(1-t), t/(1-t)] / (1-t)

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.

Note also apparently P(4,t) / (1-t)^3 = Ward Poly(4, t/(1-t)) = essentially an e.g.f. for A093500.

The compositional inverse of f(x,t) = exp[P(.,t)*x] about x=0 is

g(x,t) = { x - [t/(1-t)^2][exp[x*(1-t)]-x*(1-t)-1] }

= 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! - ... .

Can apply A134685 to these coefficients to generate f(x,t). (End)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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.

(End)

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.

Martin Klazar, Twelve countings with rooted plane trees, Europ. J. Combinatorics, Vol. 18, 1997, 195-210.

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]

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]

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

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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)

(End)

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

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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;

(End)

(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.

Sequence in context: A136230 A004732 A011244 this_sequence A142336 A114193 A039683

Adjacent sequences: A008514 A008515 A008516 this_sequence A008518 A008519 A008520

nonn,tabl,nice,easy

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

More terms from Michael Somos, Oct 13, 2002