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A112486
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Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
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12
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| The k-th diagonal of |A008275| appears as k-th column in |A008276| with k-1 leading zeros.
The recurrence, given below, is derived from diff(g1(k,x),x) - g1(k,x)= x*diff(g1(k-1,x),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.
The column sequences start with A000142 (factorials), A001705, A112487- A112491, for m=0,...,5.
The main diagonal gives (2*k-1)!! = A001147(k), k>=1.
This computation was inspired by the preprint arXiv:math-ph/0509008 v1 5 Sep 2005 by C. M. Bender, D. C. Brody and B. K. Meister: "Bernoulli-like polynomials associated with Stirling Numbers", where the Stirling polynomials are discussed.
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(a(k,m)*(x^(k-1+m))/(k-1+m)!,m=0..k-1).
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
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LINKS
| W. Lang, First 10 rows.
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FORMULA
| a(k, m)= (k+m)*a(k-1, m)+(k+m-1)*a(k-1, m-1) for m>=k>=0, a(0, 0)=1, a(k, -1):=0, a(k, m)=0 if k<m.
Contribution 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 to infin) 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)
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
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EXAMPLE
| [1]; [1,1]; [2,5,3]; [6,26,35,15]; [24,154,340,315,105]; ...
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!).
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CROSSREFS
| Cf. A112007 (triangle for o.g.f.s for unsigned Stirling1 diagonals).
Sequence in context: A024871 A111202 A163362 * A141410 A181184 A078383
Adjacent sequences: A112483 A112484 A112485 * A112487 A112488 A112489
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005
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