A049459 Generalized Stirling number triangle of first kind.

1, -4, 1, 20, -9, 1, -120, 74, -15, 1, 840, -638, 179, -22, 1, -6720, 5944, -2070, 355, -30, 1, 60480, -60216, 24574, -5265, 625, -39, 1, -604800, 662640, -305956, 77224, -11515, 1015, -49, 1, 6652800, -7893840, 4028156, -1155420, 203889

Offset: 0

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

a(n,m)= ^4P_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-(4+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 for (exp(4*t),exp(t)-1).

See A143493 for the unsigned version of this array and A143496 for the inverse. - Peter Bala, Aug 25 2008

Examples

			   1;
  -4,    1;
  20,   -9,   1;
-120,   74, -15,   1;
840, -638, 179, -22, 1;
		

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.

Links

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

Crossrefs

Unsigned column sequences are: A001715-A001719. Cf. A008275 (Stirling1 triangle), A049458, A049460. Row sums (signed triangle): A001710(n+2)*(-1)^n. Row sums (unsigned triangle): A001720(n+4).

Cf. A000035 A084938.

A143493, A143496. - Peter Bala, Aug 25 2008

Programs

Haskell

a049459 n k = a049459_tabl !! n !! k
a049459_row n = a049459_tabl !! n
a049459_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 4)
-- Reinhard Zumkeller, Mar 11 2014

Maple

A049459_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+4, n)), x, k), k=0..n): seq(print(A049459_row(n)),n=0..8); # Peter Luschny, May 16 2013

Mathematica

a[n_, m_] /; 0 <= m <= n := a[n, m] = a[n-1, m-1] - (n+3)*a[n-1, m];
a[n_, m_] /; n < m = 0;
a[_, -1] = 0; a[0, 0] = 1;
Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

Formula

a(n, m)= a(n-1, m-1) - (n+3)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

Triangle (signed) = [ -4, -1, -5, -2, -6, -3, -7, -4, -8, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [4, 1, 5, 2, 6, 3, 7, 4, 8, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143493).

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,4), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Extensions

Second formula corrected by Philippe Deléham, Nov 09 2008

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