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A049444
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Generalized Stirling number triangle of first kind.
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13
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1, -2, 1, 6, -5, 1, -24, 26, -9, 1, 120, -154, 71, -14, 1, -720, 1044, -580, 155, -20, 1, 5040, -8028, 5104, -1665, 295, -27, 1, -40320, 69264, -48860, 18424, -4025, 511, -35, 1, 362880, -663696, 509004, -214676, 54649, -8624, 826, -44, 1, -3628800, 6999840, -5753736
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n,m)= ^2P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(2+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer polynomials for (exp(2*t),exp(t)-1).
See A143491 for the unsigned version of this array and A143494 for the inverse. [From Peter Bala, Aug 25 2008]
Corresponding to the generalized Stirling number triangle of second kind A137650. - Peter Luschny, Sep 18 2011
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REFERENCES
| Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
Y. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Math. Soc. Colloquium Publications Vol. 47, 1999
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LINKS
| E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces (see pg. 27)
Y. Manin, Generating functions in algebraic geometry and sums over trees
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FORMULA
| a(n, m)= a(n-1, m-1) -(n+1)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; a(n, -1) := 0, a(0, 0)=1.
E.g.f. for m-th column of signed triangle: ((ln(1+x))^m)/(m!*(1+x)^2).
Triangle (signed) = [ -2, -1, -3, -2, -4, -3, -5, -4, -6, -5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...], where DELTA is Deleham's operator defined in A084938 (unsigned version in A143491).
E.g.f.: (1+y)^(x-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 17 2004
With P(n,t) = sum(j=0,...,n-2) a(n-2,j) * t^j and P(1,t) = -1 and P(0,t) = 1, then G(x,t) = -1 + exp[P(.,t)*x] = [(1+x)^t - 1 - t^2 * x] / [t(t-1)], whose compositional inverse in x about 0 is given in A074060. G(x,0) = -ln(1+x) and G(x,1) = (1+x) ln(1+x) - 2x . G(x,q^2) occurs in formulae on pages 194-196 of the Manin reference. - Tom Copeland, Feb 17 2008
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]
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EXAMPLE
| First few rows of the triangle are:
1
-2, 1
6, -5, 1
-24, 26, -9, 1
120, -154, 71, -14, 1
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MAPLE
| A049444_row := proc(n) local k, i;
add(add(combinat[stirling1](n, n-i), i=0..k)*x^(n-k-1), k=0..n-1);
seq(coeff(%, x, k), k=1..n-1) end:
seq(print(A049444_row(n)), n=1..7); # Peter Luschny, Sep 18 2011
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MATHEMATICA
| t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*(k+1)!*StirlingS1[n-k, i], {k, 0, n-i}]; Flatten[Table[t[n, i], {n, 0, 9}, {i, 0, n}]] [[1 ;; 48]]
(* From Jean-François Alcover, Apr 29 2011, after M. Janjic *)
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CROSSREFS
| Unsigned column sequences are A000142(n+1), A001705-A001709. Row sums (signed triangle): n!*(-1)^n, row sums (unsigned triangle): A001710(n-2). Cf. A008275 (Stirling1 triangle).
Cf. A000035 A084938, A094645, A094646.
Cf. A143491, A143494. [From Peter Bala, Aug 25 2008]
Cf. A137650.
Sequence in context: A179456 A121576 A121575 * A136124 A143491 A070918
Adjacent sequences: A049441 A049442 A049443 * A049445 A049446 A049447
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KEYWORD
| sign,easy,tabl,nice
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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EXTENSIONS
| Corrected second formula. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 09 2008
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