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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 reference.

a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n) = product(x-4*j,j=0..n-1), n >= 1, 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

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

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

Richell O. Celeste, Roberto B. Corcino, Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

W. Lang, First 10 rows.

FORMULA

a(n, m) = a(n-1, m-1) - 4*(n-1)*a(n-1, m), n >= m >= 1;

a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1.

E.g.f. for m-th column of 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

{   1};

{  -4,   1};

{  32, -12,   1};

{-384, 176, -24, 1};

...

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)

# The function bell_transform is defined in 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), for b=1, A039683 for b=2. Cf. A051141.

Cf. 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 January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)