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A112486 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.

%I A112486 #66 Apr 29 2025 18:57:08
%S A112486 1,1,1,2,5,3,6,26,35,15,24,154,340,315,105,120,1044,3304,4900,3465,
%T A112486 945,720,8028,33740,70532,78750,45045,10395,5040,69264,367884,1008980,
%U A112486 1571570,1406790,675675,135135,40320,663696,4302216,14777620,29957620
%N A112486 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
%C A112486 The k-th diagonal of |A008275| appears as the k-th column in |A008276| with k-1 leading zeros.
%C A112486 The recurrence, given below, is derived from (d/dx)g1(k,x) - g1(k,x)= x*(d/dx)g1(k-1,x) + g1(k-1,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.
%C A112486 The column sequences start with A000142 (factorials), A001705, A112487- A112491, for m=0,...,5.
%C A112486 The main diagonal gives (2*k-1)!! = A001147(k), k >= 1.
%C A112486 This computation was inspired by the Bender article (see links), where the Stirling polynomials are discussed.
%C A112486 The e.g.f. for the k-th diagonal, k >= 1, of the unsigned Stirling1 triangle |A008275| with k-1 leading zeros is g1(k-1,x) = exp(x)*Sum_{m=0..k-1} a(k,m)*(x^(k-1+m))/(k-1+m)!.
%C A112486 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
%H A112486 G. C. Greubel, <a href="/A112486/b112486.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A112486 Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%H A112486 C. M. Bender, D. C. Brody and B. K. Meister, <a href="https://arxiv.org/abs/math-ph/0509008">Bernoulli-like polynomials associated with Stirling Numbers</a>, arXiv:math-ph/0509008 [math-ph], 2005.
%H A112486 Wolfdieter Lang, <a href="/A112486/a112486.txt">First 10 rows</a>.
%F A112486 a(k, m) = (k+m)*a(k-1, m) + (k+m-1)*a(k-1, m-1) for k >= m >= 0, a(0, 0)=1, a(k, -1):=0, a(k, m)=0 if k < m.
%F A112486 From _Tom Copeland_, Oct 05 2011: (Start)
%F A112486 With polynomials
%F A112486 P(0,t) = 0
%F A112486 P(1,t) = 1
%F A112486 P(2,t) = -(1 + t)
%F A112486 P(3,t) = 2 + 5 t + 3 t^2
%F A112486 P(4,t) = -( 6 + 26 t + 35 t^2 + 15 t^3)
%F A112486 P(5,t) = 24 + 154 t +340 t^2 + 315 t^3 + 105 t^4
%F A112486 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
%F A112486   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.
%F A112486 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
%F A112486 (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).
%F A112486 The e.g.f. A(x,t) = -v * Sum_{j>=1} D(j-1,u) (-z)^j / j! where u=-(x+t+1)/t, v=1+u, z=(1+t*v)/(t*v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = -1/[t*(v-A)].(End)
%F A112486 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
%F A112486 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
%F A112486 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
%F A112486 Conjecture: row polynomials are R(n,x) = Sum_{i=0..n} Sum_{j=0..i} Sum_{k=0..j} (n+i)!*Stirling2(n+j-k,j-k)*x^k*(x+1)^(j-k)*(-1)^(n+j+k)/((n+j-k)!*(i-j)!*k!). - _Mikhail Kurkov_, Apr 21 2025
%e A112486 Triangle begins:
%e A112486     1;
%e A112486     1,    1;
%e A112486     2,    5,     3;
%e A112486     6,   26,    35,    15;
%e A112486    24,  154,   340,   315,   105;
%e A112486   120, 1044,  3304,  4900,  3465,   945;
%e A112486   720, 8028, 33740, 70532, 78750, 45045, 10395;
%e A112486 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!).
%p A112486 A112486 := proc(n,k)
%p A112486     if n < 0 or k<0 or  k> n then
%p A112486         0 ;
%p A112486     elif n = 0 then
%p A112486         1 ;
%p A112486     else
%p A112486         (n+k)*procname(n-1,k)+(n+k-1)*procname(n-1,k-1) ;
%p A112486     end if;
%p A112486 end proc: # _R. J. Mathar_, Dec 19 2013
%t A112486 A112486 [n_, k_] := A112486[n, k] = Which[n<0 || k<0 || k>n, 0, n == 0, 1, True, (n+k)*A112486[n-1, k]+(n+k-1)*A112486[n-1, k-1]]; Table[A112486[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 05 2014, after _R. J. Mathar_ *)
%Y A112486 Cf. A112007 (triangle for o.g.f.s for unsigned Stirling1 diagonals). A112487 (row sums).
%K A112486 nonn,easy,tabl
%O A112486 0,4
%A A112486 _Wolfdieter Lang_, Sep 12 2005