

A112486


Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.


12



1, 1, 1, 2, 5, 3, 6, 26, 35, 15, 24, 154, 340, 315, 105, 120, 1044, 3304, 4900, 3465, 945, 720, 8028, 33740, 70532, 78750, 45045, 10395, 5040, 69264, 367884, 1008980, 1571570, 1406790, 675675, 135135, 40320, 663696, 4302216, 14777620, 29957620
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OFFSET

0,4


COMMENTS

The kth diagonal of A008275 appears as the kth column in A008276 with k1 leading zeros.
The recurrence, given below, is derived from (d/dx)g1(k,x)  g1(k,x)= x*(d/dx)g1(k1,x) + g1(k1,x), k >= 1, with input g(1,x):=0 and initial condition g1(k,0)=1, k >= 0. This differential recurrence for the e.g.f. g1(k,x) follows from the one for unsigned Stirling1 numbers.
The column sequences start with A000142 (factorials), A001705, A112487 A112491, for m=0,...,5.
The main diagonal gives (2*k1)!! = A001147(k), k >= 1.
This computation was inspired by the Bender article (see links), where the Stirling polynomials are discussed.
The e.g.f. for the kth diagonal, k >= 1, of the unsigned Stirling1 triangle A008275 with k1 leading zeros is g1(k1,x) = exp(x)*Sum_{m=0..k1} a(k,m)*(x^(k1+m))/(k1+m)!.
a(k,n) = number of lists with entries from [n] such that (i) each element of [n] occurs at least once and at most twice, (ii) for each i that occurs twice, all entries between the two occurrences of i are > i, and (iii) exactly k elements of [n] occur twice. Example: a(1,2)=5 counts 112, 121, 122, 211, 221, and a(2,2)=3 counts 1122,1221,2211.  David Callan, Nov 21 2011


LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7.  From N. J. A. Sloane, Feb 06 2013
C. M. Bender, D. C. Brody and B. K. Meister, Bernoullilike polynomials associated with Stirling Numbers, arXiv:mathph/0509008 [mathph], 2005.
W. Lang, First 10 rows.


FORMULA

a(k, m) = (k+m)*a(k1, m) + (k+m1)*a(k1, m1) for k >= m >= 0, a(0, 0)=1, a(k, 1):=0, a(k, m)=0 if k < m.
From Tom Copeland, Oct 05 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = (1 + t)
P(3,t) = 2 + 5 t + 3 t^2
P(4,t) = ( 6 + 26 t + 35 t^2 + 15 t^3)
P(5,t) = 24 + 154 t +340 t^2 + 315 t^3 + 105 t^4
Apparently, P(n,t) = (1)^(n+1) PW[n,(1+t)] where PW are the Ward polynomials A134991. If so, an e.g.f. for the polynomials is
A(x,t) = (x+t+1)/t  LW{((t+1)/t) exp[(x+t+1)/t]}, where LW(x) is a suitable branch of the Lambert W Fct. (e.g., see A135338). The comp. inverse in x (about x = 0) is B(x) = x + (t+1) [exp(x)  x  1]. See A112487 for special case t = 1. These results are a special case of A134685 with u(x) = B(x), i.e., u_1=1 and (u_n)=(1+t) for n>0.
Let h(x,t) = 1/(dB(x)/dx) = 1/[1+(1+t)*(exp(x)1)], an e.g.f. in x for row polynomials in t of signed A028246 , then P(n,t), is given by
(h(x,t)*d/dx)^n x, evaluated at x=0, i.e., A(x,t)=exp(x*h(u,t)*d/du) u, evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t).
The e.g.f. A(x,t) = v * Sum_{j>=1} D(j1,u) (z)^j / j! where u=(x+t+1)/t, v=1+u, z=(1+t*v)/(t*v^2) and D(j1,u) are the polynomials of A042977. dA/dx = 1/[t*(vA)].(End)
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=t+1, and (a_n)=t*P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 =0.  Tom Copeland, Oct 08 2011
The row polynomials R(n,x) may be calculated using R(n,x) = 1/x^(n+1)*D^n(x), where D is the operator (x^2+x^3)*d/dx.  Peter Bala, Jul 23 2012
For n>0, Sum_{k=0..n} a(n,k)*(1/(1+W(t)))^(n+k+1) = (t d/dt)^(n+1) W(t), where W(t) is Lambert W function. For t=x, this gives Sum_{k>=1} k^(k+n)*x^k/k! =  Sum_{k=0..n} a(n,k)*(1/(1+W(x)))^(n+k+1).  Max Alekseyev, Nov 21 2019


EXAMPLE

1;
1, 1;
2, 5, 3;
6, 26, 35, 15;
24, 154, 340, 315, 105;
120, 1044, 3304, 4900, 3465, 945;
720, 8028, 33740, 70532, 78750, 45045, 10395;
k=3 column of A008276 is [0,0,2,11,35,85,175,...] (see A000914), its e.g.f. exp(x)*(2*x^2/2! + 5* x^3/3! + 3*x^4/4!).


MAPLE

A112486 := proc(n, k)
if n < 0 or k<0 or k> n then
0 ;
elif n = 0 then
1 ;
else
(n+k)*procname(n1, k)+(n+k1)*procname(n1, k1) ;
end if;
end proc: # R. J. Mathar, Dec 19 2013


MATHEMATICA

A112486 [n_, k_] := A112486[n, k] = Which[n<0  k<0  k>n, 0, n == 0, 1, True, (n+k)*A112486[n1, k]+(n+k1)*A112486[n1, k1]]; Table[A112486[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Mar 05 2014, after R. J. Mathar *)


CROSSREFS

Cf. A112007 (triangle for o.g.f.s for unsigned Stirling1 diagonals). A112487 (row sums).
Sequence in context: A163362 A243061 A242911 * A253924 A141410 A257906
Adjacent sequences: A112483 A112484 A112485 * A112487 A112488 A112489


KEYWORD

nonn,easy,tabl


AUTHOR

Wolfdieter Lang, Sep 12 2005


STATUS

approved



