A062140 - OEIS

%I #19 Mar 17 2021 04:18:18

%S 1,5,-1,30,-12,1,210,-126,21,-1,1680,-1344,336,-32,1,15120,-15120,

%T 5040,-720,45,-1,151200,-181440,75600,-14400,1350,-60,1,1663200,

%U -2328480,1164240,-277200,34650,-2310,77,-1,19958400,-31933440

%N Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).

%C The row polynomials s(n,x) := n!*L(n,4,x)= sum(a(n,m)*x^m,m=0..n) have g.f. exp(-z*x/(1-z))/(1-z)^5. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(|A008297(n,m)|*(-x)^m, m=1..n) and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

%H Indranil Ghosh, <a href="/A062140/b062140.txt">Rows 0..125</a>

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

%F T(n, m) = ((-1)^m)*n!*binomial(n+4, n-m)/m!.

%F E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^5), m >= 0.

%e Triangle begins:

%e   {1};

%e   {5,-1};

%e   {30,-12,1};

%e   {210,-126,21,-1};

%e   ...

%e 2!*L(2,4,x)=30-12*x+x^2.

%t Flatten[Table[((-1)^m)*n!*Binomial[n+4,n-m]/m!,{n,0,11},{m,0,n}]] (* _Indranil Ghosh_, Feb 23 2017 *)

%o (Python)

%o import math

%o f=math.factorial

%o def C(n,r):

%o     return f(n)//f(r)//f(n-r)

%o i=0

%o for n in range(26):

%o     for m in range(n+1):

%o         print(i, (-1)**m*f(n)*C(n+4,n-m)//f(m))

%o         i+=1 # _Indranil Ghosh_, Feb 23 2017

%o (PARI) row(n) = Vecrev(n!*pollaguerre(n, 4)); \\ _Michel Marcus_, Feb 06 2021

%Y For m=0..5 the (unsigned) columns give A001720(n+4), A062199, A062260-A062263. The row sums (signed) give A062265, the row sums (unsigned) give A062266.

%Y Cf. A021009, A062137-A062139, A066667.

%K sign,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001