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A049444 Generalized Stirling number triangle of first kind.

%I A049444 #104 Aug 11 2024 04:23:54
%S A049444 1,-2,1,6,-5,1,-24,26,-9,1,120,-154,71,-14,1,-720,1044,-580,155,-20,1,
%T A049444 5040,-8028,5104,-1665,295,-27,1,-40320,69264,-48860,18424,-4025,511,
%U A049444 -35,1,362880,-663696,509004,-214676,54649,-8624,826,-44,1,-3628800,6999840,-5753736
%N A049444 Generalized Stirling number triangle of first kind.
%C A049444 T(n, k) = ^2P_n^k in the notation of the given reference with T(0, 0) := 1. The monic row polynomials s(n,x) := Sum_{m=0..n} T(n, k)*x^k which are s(n, x) = Product_{j=0..n-1} (x-(2+j)), 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.
%C A049444 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). This translates to the usual exponential Riordan (Sheffer) notation (1/(1+x)^2, log(1+x)).
%C A049444 See A143491 for the unsigned version of this array and A143494 for the inverse. - _Peter Bala_, Aug 25 2008
%C A049444 Corresponding to the generalized Stirling number triangle of second kind A137650. - _Peter Luschny_, Sep 18 2011
%C A049444 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
%C A049444 The row polynomials s(n, x) = (-1)^n*risingfactorial(2 - x, n) are related to the column sequences of the unsigned Abel triangle A137452(n, k), for k >= 2. See the formula there. - _Wolfdieter Lang_, Nov 21 2022
%D A049444 Y. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Math. Soc. Colloquium Publications Vol. 47, 1999. [From _Tom Copeland_, Jun 29 2008]
%D A049444 S. Roman, The Umbral Calculus, Academic Press, 1984 (also Dover Publications, 2005).
%H A049444 Reinhard Zumkeller, <a href="/A049444/b049444.txt">Rows n = 0..125 of triangle, flattened</a>
%H A049444 E. Getzler, <a href="http://arxiv.org/abs/alg-geom/9411004">Operads and moduli spaces of genus 0 Riemann surfaces</a>, arXiv:alg-geom/9411004, 1994, (see p. 23, g(x,t)). [From _Tom Copeland_, Dec 11 2011]
%H A049444 T. Hyde and J. Lagarias <a href="http://arxiv.org/abs/1604.05359">Polynomial splitting measures and cohomology of the pure braid group</a>, arXiv preprint arXiv:1604.05359 [math.RT], 2016.
%H A049444 Y. Manin, <a href="http://arxiv.org/abs/alg-geom/9407005">Generating functions in algebraic geometry and sums over trees</a>, arXiv:alg-geom/9407005, 1994, (Eqn. 0.7 and 1.7). [From _Tom Copeland_, Dec 10 2011]
%H A049444 D. S. Mitrinovic and M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
%H A049444 R. Murri, <a href="http://arxiv.org/abs/1202.1820">Fatgraph Algorithms and the Homology of the Kontsevich Complex</a>, arXiv:1202.1820 [math.AG], 2012, (see Table 1, p. 3). [From _Tom Copeland_, Sep 18 2012]
%F A049444 T(n, k) = T(n-1, k-1) - (n+1)*T(n-1, k), n >= k >= 0; T(n, k) = 0, n < k; T(n, -1) = 0, T(0, 0) = 1.
%F A049444 E.g.f. for k-th column of signed triangle: ((log(1+x))^k)/(k!*(1+x)^2).
%F A049444 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).
%F A049444 E.g.f.: (1 + x)^(y-2). - _Vladeta Jovovic_, May 17 2004 [For row polynomials s(n, y)]
%F A049444 With P(n, t) = Sum_{j=0..n-2} T(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
%F A049444 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
%F A049444 T(n, k) = Sum_{j=0..n} (-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k). - _Mélika Tebni_, May 02 2022
%F A049444 From _Wolfdieter Lang_, Nov 24 2022: (Start)
%F A049444 Recurrence for row polynomials {s(n, x)}_{n>=0}: s(0, x) = 1, s(n, x) = (x - 2)*exp(-(d/dx)) s(n-1, x), for n >= 1. This is adapted from the general Sheffer result given by S. Roman, Corollary 3.7.2., p. 50.
%F A049444 Recurrence for column sequence {T(n, k)}_{n>=k}: T(n, n) = 1, T(n, k) = (n!/(n-k))*Sum_{j=k..n-1} (1/j!)*(a(n-1-j) + k*beta(n-1-j))*T(n-1, k), for k >= 0, where alpha = repeat(-2, 2) and beta(n) = [x^n] (d/dx)log(log(x)/x) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), for n >= 0. This is the adapted Boas-Buck recurrence, also given in Rainville, Theorem 50., p. 141, For the references and a comment see A046521. (End)
%e A049444 The Triangle  begins:
%e A049444 n\k       0       1        2       3       4      5      6    7   8 9 ...
%e A049444 0:        1
%e A049444 1:       -2       1
%e A049444 2:        6      -5        1
%e A049444 3:      -24      26       -9       1
%e A049444 4:      120    -154       71     -14       1
%e A049444 5      -720    1044     -580     155     -20      1
%e A049444 6:     5040   -8028     5104   -1665     295    -27      1
%e A049444 7:   -40320   69264   -48860   18424   -4025    511    -35    1
%e A049444 8:   362880 -663696   509004 -214676   54649  -8624    826  -44
%e A049444 9: -3628800 6999840 -5753736 2655764 -761166 140889 -16884 1266 -54 1
%e A049444 ...  [reformatted by _Wolfdieter Lang_, Nov 21 2022]
%p A049444 A049444_row := proc(n) local k,i;
%p A049444 add(add(Stirling1(n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
%p A049444 seq(coeff(%,x,k),k=1..n-1) end:
%p A049444 seq(print(A049444_row(n)),n=1..7); # _Peter Luschny_, Sep 18 2011
%p A049444 A049444:= (n, k)-> add((-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k), j=0..n):
%p A049444 seq(print(seq(A049444(n, k), k=0..n)), n=0..11);  # _Mélika Tebni_, May 02 2022
%t A049444 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]]
%t A049444 (* _Jean-François Alcover_, Apr 29 2011, after _Milan Janjic_ *)
%o A049444 (Haskell)
%o A049444 a049444 n k = a049444_tabl !! n !! k
%o A049444 a049444_row n = a049444_tabl !! n
%o A049444 a049444_tabl = map fst $ iterate (\(row, i) ->
%o A049444    (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 2)
%o A049444 -- _Reinhard Zumkeller_, Mar 11 2014
%Y A049444 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).
%Y A049444 Cf. A000035, A084938, A094645, A094646.
%Y A049444 Cf. A143491, A143494.
%Y A049444 Cf. A136124, A137452, A137650.
%K A049444 sign,easy,tabl,nice
%O A049444 0,2
%A A049444 _Wolfdieter Lang_
%E A049444 Second formula corrected by _Philippe Deléham_, Nov 09 2008