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A051142 Generalized Stirling number triangle of first kind. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..39.

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].

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).

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A123126 A303277 A174501 * A266240 A322601 A075804

Adjacent sequences:  A051139 A051140 A051141 * A051143 A051144 A051145

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified May 9 00:09 EDT 2021. Contains 343685 sequences. (Running on oeis4.)