cp's OEIS Frontend

A289215 a(n) = n! * Laguerre(n,-9).

%I A289215 #29 Feb 16 2025 08:33:48
%S A289215 1,10,119,1626,24945,422994,7836255,157169826,3388099329,78031713690,
%T A289215 1910451937671,49510386761130,1353167691897969,38878205830928226,
%U A289215 1170930069982659375,36875214316479123954,1211549306913066598785,41445016025330141416746
%N A289215 a(n) = n! * Laguerre(n,-9).
%H A289215 Alois P. Heinz, <a href="/A289215/b289215.txt">Table of n, a(n) for n = 0..430</a>
%H A289215 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A289215 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A289215 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A289215 E.g.f.: exp(9*x/(1-x))/(1-x).
%F A289215 a(n) = n! * Sum_{i=0..n} 9^i/i! * binomial(n,i).
%F A289215 a(n) = n! * A160601(n)/A160602(n).
%F A289215 a(n) ~ exp(-9/2 + 6*sqrt(n) - n) * n^(n + 1/4) / sqrt(6) * (1 + 181/(48*sqrt(n))). - _Vaclav Kotesovec_, Nov 13 2017
%F A289215 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 9^n * x^n / (n!)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%p A289215 a:= n-> n! * add(binomial(n, i)*9^i/i!, i=0..n):
%p A289215 seq(a(n), n=0..20);
%t A289215 Table[n!*LaguerreL[n, -9], {n, 0, 20}] (* _Indranil Ghosh_, Jul 04 2017 *)
%o A289215 (Python)
%o A289215 from mpmath import *
%o A289215 mp.dps=100
%o A289215 def a(n): return int(fac(n)*laguerre(n, 0, -9))
%o A289215 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Jul 04 2017
%o A289215 (PARI) my(x = 'x + O('x^30)); Vec(serlaplace(exp(9*x/(1-x))/(1-x))) \\ _Michel Marcus_, Jul 04 2017
%o A289215 (PARI) a(n) = n!*pollaguerre(n, 0, -9); \\ _Michel Marcus_, Feb 05 2021
%o A289215 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(9*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 11 2018
%Y A289215 Column k=9 of A289192.
%Y A289215 Cf. A160601, A160602.
%K A289215 nonn
%O A289215 0,2
%A A289215 _Alois P. Heinz_, Jun 28 2017