OFFSET
0,3
COMMENTS
From Peter Luschny, Jan 17 2023: (Start)
a(n) is the number of connection patterns in a telephone system with n possibilities of connection and n subscribers.
The number of matchings of a complete multigraph K(n, n).
The main diagonal of A359762. (End)
Let k be a positive integer. It appears that reducing this sequence modulo k produces an eventually periodic sequence. For example, modulo 10 the sequence becomes [1, 3, 0, 3, 6, 1, 8, 5, 8, 1, 6, 3, 0, 3, 6, 1, 8, 5, 8, 1, 6, 3, 0, 3, 6, ...], with an apparent period [1, 8, 5, 8, 1, 6, 3, 0, 3, 6] of length 10 starting at a(5). - Peter Bala, Apr 16 2023
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..416
Urszula Bednarz and Małgorzata Wołowiec-Musiał, On a new generalization of telephone numbers, Turkish Journal of Mathematics: Vol. 43: No. 3, (2019).
FORMULA
E.g.f.: exp( sqrt(-LambertW(-x^2)) ) / (1 + LambertW(-x^2)).
a(n) ~ (exp(1) + (-1)^n*exp(-1)) * n^n / (sqrt(2) * exp(n/2)). - Vaclav Kotesovec, Nov 11 2016
a(n) = Sum_{j=0..n, j even} binomial(n, j) * (j - 1)!! * n^(j/2). - Peter Luschny, Jan 17 2023
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 73*x^4/4! + 426*x^5/5! + 4951*x^6/6! + 41308*x^7/7! + 658785*x^8/8! + 7149628*x^9/9! + 144963451*x^10/10! + ...
The table of coefficients of x^k/k! in exp(x + n*x^2/2) begins:
n=0: 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
n=1: 1, 1, 2, 4, 10, 26, 76, 232, 764, ...;
n=2: 1, 1, 3, 7, 25, 81, 331, 1303, 5937, ...;
n=3: 1, 1, 4, 10, 46, 166, 856, 3844, 21820, ...;
n=4: 1, 1, 5, 13, 73, 281, 1741, 8485, 57233, ...;
n=5: 1, 1, 6, 16, 106, 426, 3076, 15856, 123516, ...;
n=6: 1, 1, 7, 19, 145, 601, 4951, 26587, 234529, ...;
n=7: 1, 1, 8, 22, 190, 806, 7456, 41308, 406652, ...;
n=8: 1, 1, 9, 25, 241, 1041, 10681, 60649, 658785, ...;
n=9: 1, 1, 10, 28, 298, 1306, 14716, 85240, 1012348, ...;
n=10:1, 1, 11, 31, 361, 1601, 19651, 115711, 1491281, ...; ...
in which the main diagonal forms this sequence.
In the above table, the e.g.f. of the m-th diagonal equals the e.g.f. of this sequence multiplied by ( LambertW(-x^2)/(-x^2) )^(m/2).
Example,
A(x)*sqrt(-LambertW(-x^2))/x = 1 + x + 4*x^2/2! + 13*x^3/3! + 106*x^4/4! + 601*x^5/5! + 7456*x^6/6! + 60649*x^7/7! + 1012348*x^8/8! + ...
equals the e.g.f. of the next lower diagonal in the table.
RELATED SERIES.
-LambertW(-x^2) = x^2 + 2*x^4/2! + 3^2*x^6/3! + 4^3*x^8/4! + 5^4*x^10/5! + 6^5*x^12/6! + ... + n^(n-1)*x^(2*n)/n! + ...
sqrt(-LambertW(-x^2)) = x + 3^0*x^3/(1!*2) + 5*x^5/(2!*2^2) + 7^2*x^7/(3!*2^3) + 9^3*x^9/(4!*2^4) + ... + (2*n+1)^(n-1)*x^(2*n+1)/(n!*2^n) + ...
MAPLE
a := n -> add(binomial(n, j) * doublefactorial(j-1) * n^(j/2), j = 0..n, 2):
seq(a(n), n = 0..25); # Peter Luschny, Jan 17 2023
PROG
(PARI) {a(n) = n!*polcoeff( exp(x + n*x^2/2 + x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(Python)
from math import factorial, comb
def oddfactorial(n: int) -> int:
return factorial(2 * n) // (2**n * factorial(n))
def a(n: int) -> int:
return sum(comb(n, 2*j) * oddfactorial(j) * n**j for j in range(n+1))
print([a(n) for n in range(26)]) # Peter Luschny, Jan 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 10 2016
STATUS
approved