A086915 - OEIS

%I #40 Sep 08 2022 08:45:11

%S 2,4,4,12,24,8,48,144,96,16,240,960,960,320,32,1440,7200,9600,4800,

%T 960,64,10080,60480,100800,67200,20160,2688,128,80640,564480,1128960,

%U 940800,376320,75264,7168,256,725760,5806080,13547520,13547520,6773760,1806336

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

%C Also the Bell transform of A052849(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016

%C 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

%H G. C. Greubel, <a href="/A086915/b086915.txt">Rows n=1..100 of the triangle, flattened</a>

%H D. Gross and A. Matytsin, <a href="http://arxiv.org/abs/hep-th/9404004">Instanton induced large N phase transitions in two and four dimensional QCD</a>, arXiv:hep-th/9404004, 1994.

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

%F E.g.f.: exp(2*x*y/(1-x)).

%F From _G. C. Greubel_, Feb 23 2021: (Start)

%F T(n, k) = (-2)^k * A008297(n, k) = 2^k * A105278(n, k).

%F 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)

%e Triangle begins:

%e    2;

%e    4,   4;

%e   12,  24,  8;

%e   48, 144, 96, 16;

%e   ...

%p # The function BellMatrix is defined in A264428.

%p # Adds (1, 0, 0, 0, ...) as column 0.

%p BellMatrix(n -> 2*(n+1)!, 9); # _Peter Luschny_, Jan 26 2016

%t Flatten[Table[n!/k! Binomial[n-1,k-1]2^k,{n,10},{k,n}]] (* _Harvey P. Dale_, May 25 2011 *)

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[2*(#+1)!&, rows = 12];

%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%o (PARI) for(n=1,10, for(k=1, n, print1(n!/k!*binomial(n-1,k-1)*2^k, ", "))) \\ _G. C. Greubel_, May 23 2018

%o (Magma) [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

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

%K nonn,tabl

%O 1,1

%A _Vladeta Jovovic_, Sep 24 2003