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A049444 Generalized Stirling number triangle of first kind. 15
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; text; internal format)
OFFSET

0,2

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_{m=0..n} a(n,m)*x^m which are s(n,x) = Product_{k=0..n-1} (x-(2+k)), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*S1(n-k,y), with the Stirling1 polynomials S1(n,x) = Sum_{m=1..n} (A008275(n,m)*x^m) 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. - Peter Bala, Aug 25 2008

Corresponding to the generalized Stirling number triangle of second kind A137650. - Peter Luschny, Sep 18 2011

Unsigned, reversed rows (cf. A145324, A136124) are the dimensions of the cohomology of a complex manifold with a symmetric group (S_n) action. See p. 17 of the Hyde and Lagarias link. See also the Murri link for an interpretation as the Betti numbers of the moduli space M(0,n) of smooth Riemann surfaces. - Tom Copeland, Dec 09 2016

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 [From Tom Copeland, Jun 29 2008]

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, arXiv:alg-geom/9411004, 1994, (see p. 23, g(x,t)). [From Tom Copeland, Dec 11 2011]

T. Hyde and J. Lagarias Polynomial splitting measures and cohomology of the pure braid group, arXiv preprint arXiv:1604.05359 [math.RT], 2016.

Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alg-geom/9407005, 1994, (Eqn. 0.7 and 1.7). [From Tom Copeland, Dec 10 2011]

R. Murri, Fatgraph Algorithms and the Homology of the Kontsevich Complex, arXiv:1202.1820 [math.AG], 2012, (see Table 1, p. 3). [From Tom Copeland, Sep 18 2012]

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: ((log(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 Deléham's operator defined in A084938 (unsigned version in A143491).

E.g.f.: (1+y)^(x-2). - Vladeta Jovovic, 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) = -log(1+x) and G(x,1) = (1+x) log(1+x) - 2x. G(x,q^2) occurs in formulas on pages 194-196 of the Manin reference. - Tom Copeland, Feb 17 2008

If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then T(n,i) = f(n,i,2), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008

EXAMPLE

First few rows of the triangle are:

              1;

          -2,    1;

        6,   -5,    1;

   -24,   26,   -9,    1;

120, -154,   71,  -14,    1;

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

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]]

(* Jean-François Alcover, Apr 29 2011, after Milan Janjic *)

PROG

(Haskell)

a049444 n k = a049444_tabl !! n !! k

a049444_row n = a049444_tabl !! n

a049444_tabl = map fst $ iterate (\(row, i) ->

   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 2)

-- Reinhard Zumkeller, Mar 11 2014

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. - Peter Bala, Aug 25 2008

Cf. A137650.

Cf. A136124.

Sequence in context: A214152 A121575 A121576 * A136124 A143491 A295517

Adjacent sequences:  A049441 A049442 A049443 * A049445 A049446 A049447

KEYWORD

sign,easy,tabl,nice

AUTHOR

Wolfdieter Lang

EXTENSIONS

Second formula corrected by Philippe Deléham, Nov 09 2008

STATUS

approved

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Last modified January 20 21:36 EST 2019. Contains 319336 sequences. (Running on oeis4.)