A051142 Generalized Stirling number triangle of first kind.

1, -4, 1, 32, -12, 1, -384, 176, -24, 1, 6144, -3200, 560, -40, 1, -122880, 70144, -14400, 1360, -60, 1, 2949120, -1806336, 415744, -47040, 2800, -84, 1, -82575360, 53526528, -13447168, 1732864, -125440, 5152, -112, 1, 2642411520, -1795424256, 483835904

Offset: 1

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

a(n,m) = R_n^m(a=0, b=4) in the notation of the given 1961 and 1962 references.

a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 4*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

This is the signed Stirling1 triangle with diagonal d >= 0 (main diagonal d = 0) scaled with 4^d.

Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+4) (A047053) giving unsigned values and adding 1, 0, 0, 0, ... as column 0. For the definition of the Bell transform, see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015

Examples

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -4,     1;
       32,   -12,      1;
     -384,   176,    -24,    1;
     6144, -3200,    560,  -40,   1,
  -122880, 70144, -14400, 1360, -60, 1;
  ...
3rd row o.g.f.: E(3,x) = 32*x - 12*x^2 + x^3.
		

Links

• Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
• Wolfdieter Lang, First 10 rows.
• D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
• D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
• D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].

Crossrefs

First (m=1) column sequence is: A047053(n-1).

Row sums (signed triangle): A008545(n-1)*(-1)^(n-1).

Row sums (unsigned triangle): A007696(n).

Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3).

Cf. A039692, A264428, A265606.

Programs

Mathematica

Table[StirlingS1[n, m] 4^(n - m), {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

Sage

# uses[bell_transform from A264428]
# Unsigned values and an additional first column (1,0,0,0, ...).
def A051142_row(n):
    multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
    mfact = [multifact_4_4(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A051142_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

Formula

a(n, m) = a(n-1, m-1) - 4*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.

E.g.f. for the m-th column of the signed triangle: (log(1 + 4*x)/4)^m/m!.

a(n, m) = S1(n, m)*4^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).