A152059 - OEIS

%I #42 Feb 22 2024 20:33:16

%S 1,2,13,136,1961,36046,805597,21204548,642451441,22021483546,

%T 842527453421,35591363004352,1645373927307673,82625931422081126,

%U 4478815087922020861,260648364396903639676,16208855884741850686817

%N a(n) is the number of ways 2n-1 seats can be occupied by at most n people for n>=1, with a(0)=1.

%C Let A(x) be the e.g.f. of this sequence, and B(x) be the e.g.f. of A082545, then B(x)/A(x) = C(x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108). This follows from the fact that this sequence and A082545 form adjacent semi-diagonals of table A088699. - _Paul D. Hanna_, Aug 16 2022

%H Seiichi Manyama, <a href="/A152059/b152059.txt">Table of n, a(n) for n = 0..366</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

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

%F a(n) = Sum_{k=0..n} k! * C(2*n-1,k) * C(n,k).

%F Central terms of triangle A086885 (after initial term).

%F a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^n. - _Ilya Gutkovskiy_, Oct 02 2017

%F a(n) ~ 2^(2*n - 1/2) * n^n / exp(n-1). - _Vaclav Kotesovec_, Oct 02 2017

%F a(n) = n! * pollaguerre(n, n-1, -1). - _Seiichi Manyama_, May 01 2021

%F From _Paul D. Hanna_, Aug 16 2022: (Start)

%F E.g.f.: exp( (1-2*x - sqrt(1-4*x))/(2*x) ) / ((sqrt(1-4*x) - (1-4*x))/(2*x)), derived from the e.g.f for A082545 given by _Mark van Hoeij_.

%F E.g.f.: exp(C(x) - 1) / (2 - C(x)), where C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). (End)

%t Table[(-1)^n * HypergeometricU[-n, n, -1], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 02 2017 *)

%o (PARI) a(n)=sum(k=0,n,k!*binomial(2*n-1, k)*binomial(n, k))

%o (PARI) a(n) = n!*pollaguerre(n, n-1, -1); \\ _Seiichi Manyama_, May 01 2021

%o (Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n, n-1), -1): n in [0..40]]; // _G. C. Greubel_, Aug 11 2022

%o (SageMath) [factorial(n)*gen_laguerre(n, n-1, -1) for n in (0..40)] # _G. C. Greubel_, Aug 11 2022

%Y Cf. A082545, A086885, A343832, A088699, A000108.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 22 2008