A079621 -id:A079621 - cp's OEIS frontend

2, 4, 4, 12, 24, 8, 48, 144, 96, 16, 240, 960, 960, 320, 32, 1440, 7200, 9600, 4800, 960, 64, 10080, 60480, 100800, 67200, 20160, 2688, 128, 80640, 564480, 1128960, 940800, 376320, 75264, 7168, 256, 725760, 5806080, 13547520, 13547520, 6773760, 1806336

Offset: 1

Vladeta Jovovic, Sep 24 2003

Also the Bell transform of A052849(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

The coefficients of n! * L_n(-2*x,-1), where n! * L_n(-x,-1) are the normalized, unsigned Laguerre polynomials of order -1 of A105278, also known as the Lah polynomials, which are also a shifted version of n! * L_n(-x,1). Cf. p. 8 of the Gross and Matytsin link. - Tom Copeland, Sep 30 2016

			Triangle begins:
   2;
   4,   4;
  12,  24,  8;
  48, 144, 96, 16;
  ...

G. C. Greubel, Rows n=1..100 of the triangle, flattened
D. Gross and A. Matytsin, Instanton induced large N phase transitions in two and four dimensional QCD, arXiv:hep-th/9404004, 1994.
Index entries for sequences related to Laguerre polynomials

Cf. A008297, A052897 (row sums), A059110, A079621, A105278.

[Factorial(n)*Binomial(n-1,k-1)*2^k/Factorial(k): k in [1..n], n in [1..10]]; // G. C. Greubel, May 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> 2*(n+1)!, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[n!/k! Binomial[n-1,k-1]2^k,{n,10},{k,n}]] (* Harvey P. Dale, May 25 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

for(n=1,10, for(k=1, n, print1(n!/k!*binomial(n-1,k-1)*2^k, ", "))) \\ G. C. Greubel, May 23 2018

E.g.f.: exp(2*x*y/(1-x)).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-2)^k * A008297(n, k) = 2^k * A105278(n, k).
Sum_{k=1..n} T(n, k) = 2 * n! * Hypergeometric1F1([1-n], [2], -2) = 2*(n-1)! * LaguerreL(n-1, 1, -2) = A253286(n, 2). (End)

cp's OEIS Frontend

A086915 Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula