A277614 a(n) is the coefficient of x^n/n! in exp(x + n*x^2/2).

1, 1, 3, 10, 73, 426, 4951, 41308, 658785, 7149628, 144963451, 1937124696, 47660873833, 756536698360, 21888570052623, 402400189738576, 13384439813823361, 279666289640774928, 10512823691028429235, 246061359639756047008, 10314843348672697017801, 267328220273408530004896, 12363686002049118477390343, 351473836594567725961268160, 17776996370247936310502612833, 550002942283550733215994429376

Offset: 0

Author

Paul D. Hanna, Nov 10 2016

Keywords

• nonn

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

Examples

			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) + ...
		

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).

Crossrefs

Cf. A359762, A362300, A362314, A362319.

Programs

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

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

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