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A094646
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Generalized Stirling number triangle of first kind.
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4
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1, -2, 1, 2, -3, 1, 0, 2, -3, 1, 0, 2, -1, -2, 1, 0, 4, 0, -5, 0, 1, 0, 12, 4, -15, -5, 3, 1, 0, 48, 28, -56, -35, 7, 7, 1, 0, 240, 188, -252, -231, 0, 42, 12, 1, 0, 1440, 1368, -1324, -1638, -231, 252, 114, 18, 1, 0, 10080, 11016, -7900, -12790, -3255, 1533, 1050, 240, 25, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Triangle T(n,k), 0<=k<=n, read by rows, given by [ -2, 1, -1, 2, 0, 3, 1, 4, 2, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 23 2006
From Wolfdieter Lang, Jun 23 2011: (Start)
The row polynomials s(n,x):=sum(T(n,k)*x^k,k=0..n) satisfy risefac(x-2,n)=s(n,x), with the rising factorials risefac(x-2,n):=product(x-2+j,j=0..n-1), n>=1, risefac(x-2,0)=1. Compare this with the formula risefac(x,n)=|S1|(n,x), with the row polynomials |S1|(n,x) of A132393 (unsigned Stirling1).
This is the third triangle of an a-family of Sheffer arrays, call them |S1|(a), with e.g.f. of the row polynomials |S1|(a;x;z) = ((1-z)^a)*exp(-x*log(1-z)).
In the notation showing the column e.g.f.s this is
Sheffer ((1-z)^a,-log(1-z)). In the umbral notation (see the Roman reference, given under A094645) this is called Sheffer for (exp(a*t),1-exp(-t)). For a=0 this becomes the unsigned Stirling1 triangle |S1|(0) = A132393 with row polynomials |S1|(0;n,x) =: s1(n,x).
E.g.f. column number k (with leading zeros): ((1-x)^a)*((-log(1-x))^k)/k!, k>=0.
E.g.f. for row sums is (1-x)^(a-1), and the e.g.f. for the alternating row sums is (1-x)^(a+1).
Row polynomial recurrence:
|S1|(a;n,x)=(x+(n-1-a))*|S1|(a;n-1,m), |S1|(a;0,x)=1.
Meixner identity (see the reference under A060338):
|S1|(a;n,x) - |S1|(a;n,x-1) = n*|S1|(a;n-1,x),n>=1,
Also (from the corollary 3.7.2 on p. 50 of the Roman reference): |S1|(a;n,x) = (x-a)*|S1|(a;n-1,x+1), n>=1.
Recurrence: |S1|(a;n,k) = |S1|(a;n-1,k-1) + (n-(a+1))*|S1|(a;n-1,k); |S1|(a;n,k)=0 if n<m, |S1|(a;n,-1)=0, |S1|(a;0,0)=1.
Connection to |Stirling1|=|S1|(0):
|S1|(a;n,k)= sum(|S1|(a;a,p)*abs(Stirling1(n-a,k-p)) ,p=0..a), n>=a.
The exponential convolution identity is
|S1|(a;n,x+y) = sum(binomial(n,k)*|S1|(a;k,y)*s1(n-k,x),k=0..n), n>=0, with symmetry x <-> y.
The Sheffer a- and z-sequences are (see the W. Lang link under A006232): Sha(a;n)=A164555(n)/A027642(n) (independent of a) with e.g.f. x/(1-exp(-x)), and the z-sequence has e.g.f.(exp(a*x)-1)/(exp(-x)-1).
The inverse Sheffer matrix has e.g.f. exp(a*z)*exp(x*(1-exp(-z))), in short notation (exp(a*z),1-exp(-z)),
(or in umbral notation ((1-t)^a,-log(1-t))).
(End)
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FORMULA
| E.g.f.: (1-y)^(2-x).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2007
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then |T(n,i)| =| f(n,i,-2)|, for n=1,2,...;i=0...n. [From Milan R. Janjic (agnus(AT)blic.net), Dec 21 2008]
From Wolfdieter Lang, Jun 23 2011: (Start)
risefac(x-2,n)=sum(T(n,k)*x^k,k=0..n),n>=0, with the rising factorials (see a comment above).
Recurrence: T(n,k) = T(n-1,k-1) + (n-3)*T(n-1,k); T(n,k)=0 if n<m, T(n,-1)=0, T(0,0)=1.
T(n,k) = 2*abs(S1(n-2,k)) - 3*abs(S1(n-2,k-1)) + abs(S1(n-2,k-2)), n>=2, with S1(n,k) = Stirling1(n,k) = A048994(n,k).
E.g.f. column number k (with leading zeros):
((1-x)^2)*((-log(1-x))^k)/k!, k>=0.
E.g.f. for row sums is 1-x, i.e., [1,-1,0,0,...],
and the e.g.f. for the alternating row sums is (1-x)^3. i.e., [1,-3,3,1,0,0,...].
(End)
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EXAMPLE
| 1;
-2, 1;
2,-3, 1;
0, 2,-3, 1;
0, 2,-1,-2, 1;
0, 4, 0,-5, 0, 1;
...
risefac(x-2,3)= (x-2)*(x-1)*x = 2*x-3*x^2+x^3.
-1 = T(4,2) = T(3,1) + 1*T(3,2) = 2 + (-3).
T(4,3) = 2*abs(S1(2,3))-3*abs(S1(2,2))+1*abs(S1(2,1))
= 2*0 - 3*1 + 1*1 = -2.
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MATHEMATICA
| Flatten[ Table[ CoefficientList[ Pochhammer[x-2, n], x], {n, 0, 10}]] (* From Jean-François Alcover, Sep 26 2011 *)
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CROSSREFS
| Cf. A049444, A049458, A094645.
Sequence in context: A082501 A132815 A167684 * A124448 A143343 A138243
Adjacent sequences: A094643 A094644 A094645 * A094647 A094648 A094649
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KEYWORD
| easy,sign,tabl
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), May 17 2004
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