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A094645
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Generalized Stirling number triangle of first kind.
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10
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1, -1, 1, 0, -1, 1, 0, -1, 0, 1, 0, -2, -1, 2, 1, 0, -6, -5, 5, 5, 1, 0, -24, -26, 15, 25, 9, 1, 0, -120, -154, 49, 140, 70, 14, 1, 0, -720, -1044, 140, 889, 560, 154, 20, 1, 0, -5040, -8028, -64, 6363, 4809, 1638, 294, 27, 1, 0, -40320, -69264, -8540, 50840, 44835, 17913, 3990, 510, 35, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,12
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COMMENTS
| From Wolfdieter Lang, Jun 20 2011: (Start)
The row polynomials s(n,x):=sum(T(n,k)*x^k,k=0..n) satisfy risefac(x-1,n)=s(n,x), with the rising factorials risefac(x-1,n):=product(x-1+j,j=0..n-1), n>=1, risefac(x-1,0)=1. Compare with the formula risefac(x,n)=s1(n,x), with the row polynomials s1(n,x) of A132393 (unsigned Stirling1).
This is the lower triangular Sheffer array with e.g.f.
T(x,z) = (1-z)*exp(-x*ln(1-z)) (the rewritten e.g.f. from the formula section). See the W. Lang link under A006232 for Sheffer matrices and the Roman reference. In the notation which indicates the column e.g.f.s this is Sheffer (1-z,-ln(1-z)). In the umbral notation (cf. Roman) this is called Sheffer for (exp(t),1-exp(-t)).
The row polynomials satisfy s(n,x) = (x+n-1)*s(n-1,x), s(0,x)=1, and s(n,x) = (x-1)*s1(n-1,x), n>=1, s1(0,x)=1, with the unsigned Stirling1 row polynomials s1(n,x).
The row polynomials satisfy also
s(n,x) - s(n,x-1) = n*s(n-1,x), n>1, s(0,x)=1.
(from the Meixner identity, see the Meixner reference given under A060338).
The row polynomials satisfy as well (from corollary 3.7.2. p. 50 of the Roman reference)
s(n,x) = (x-1)*s(n-1,x+1), n>=1, s(0,n)=1.
The exponential convolution identity is
s(n,x+y) = sum(binomial(n,k)*s(k,y)*s1(n-k,x),k=0..n),
n>=0, with symmetry x <-> y.
The row sums are 1 for n=0 and 0 else, and the alternating row sums are 1,-2,2, followed by zeros, with e.g.f.(1-x)^2.
The Sheffer a-sequence Sha(n)=A164555(n)/A027642(n) with e.g.f. x/(1-exp(-x)), and the z-sequence is Shz(n)=-1 with e.g.f. -exp(x).
The inverse Sheffer matrix is ((-1)^(n-k))*A105794(n,k) with e.g.f. exp(z)*exp(x*(1-exp(-z))). (End)
Triangle T(n,k), read by rows, given by (-1, 1, 0, 2, 1, 3, 2, 4, 3, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Jan 16 2012
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REFERENCES
| S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
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FORMULA
| E.g.f.: (1-y)^(1-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 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 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,-1)|, for n=1,2,...;i=0...n. [From Milan R. Janjic (agnus(AT)blic.net), Dec 21 2008]
From Wolfdieter Lang, Jun 20 2011: (Start)
T(n,k) = |S1(n-1,k-1)| - |S1(n-1,k)|, n>=1, k>=1, with |S1(n,k)|= A132393(n,k) (unsigned Stirling1).
Recurrence: T(n,k) = T(n-1,k-1) +(n-2)*T(n-1,k) if n>=k>=0; T(n,k)=0 if n<k; T(n,-1)=0; T(0,0)=1.
E.g.f. column k: (1-x)*((-ln(1-x))^k)/k!. (End)
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EXAMPLE
| 1;
-1, 1;
0, -1, 1;
0, -1, 0, 1;
0, -2, -1, 2, 1;
0, -6, -5, 5, 5, 1;
0,-24,-26, 15, 25, 9, 1;
...
Recurrence:
-2 = T(4,1) = T(3,0) + (4-2)*T(3,1) = 0 + 2*(-1).
Row polynomials:
s(3,x) = -x+x^3 = (x-1)*s1(2,x) = (x-1)*(x+x^2).
s(3,x) = (x-1)*s(2,x+1) = (x-1)*(-(x+1)+(x+1)^2).
s(3,x) - s(3,x-1) = -x+x^3 -(-(x-1)+(x-1)^3)
= 3*(-x+x^2) = 3*s(2,x).
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MATHEMATICA
| t[n_, k_] /; n >= k >= 0 := t[n, k] = t[n-1, k-1] + (n-2)*t[n-1, k]; t[n_, k_] /; n < k = 0; t[_, -1] = 0; t[0, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, 10}, {k, 0, n}] ] (* From Jean-François Alcover, Sep 29 2011, after recurrence *)
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CROSSREFS
| Cf. A049444, A049458, A094646, A132393, A105794.
Sequence in context: A079532 A191312 * A105793 A158566 A128410 A059782
Adjacent sequences: A094642 A094643 A094644 * A094646 A094647 A094648
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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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