A152059 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.

1, 2, 13, 136, 1961, 36046, 805597, 21204548, 642451441, 22021483546, 842527453421, 35591363004352, 1645373927307673, 82625931422081126, 4478815087922020861, 260648364396903639676, 16208855884741850686817

Offset: 0

Author

Paul D. Hanna, Nov 22 2008

Keywords

• nonn

Comments

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..366
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

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

Programs

Magma

[Factorial(n)*Evaluate(LaguerrePolynomial(n, n-1), -1): n in [0..40]]; // G. C. Greubel, Aug 11 2022

Mathematica

Table[(-1)^n * HypergeometricU[-n, n, -1], {n, 0, 20}] (* Vaclav Kotesovec, Oct 02 2017 *)

PARI

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

PARI

a(n) = n!*pollaguerre(n, n-1, -1); \\ Seiichi Manyama, May 01 2021

SageMath

[factorial(n)*gen_laguerre(n, n-1, -1) for n in (0..40)] # G. C. Greubel, Aug 11 2022

Formula

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

Central terms of triangle A086885 (after initial term).

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^n. - Ilya Gutkovskiy, Oct 02 2017

a(n) ~ 2^(2*n - 1/2) * n^n / exp(n-1). - Vaclav Kotesovec, Oct 02 2017

a(n) = n! * pollaguerre(n, n-1, -1). - Seiichi Manyama, May 01 2021

From Paul D. Hanna, Aug 16 2022: (Start)

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.

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)