A362314 -id:A362314 - cp's OEIS frontend

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

Paul D. Hanna, Nov 10 2016

nonn

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

			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) + ...

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).

Cf. A359762, A362300, A362314, A362319.

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

{a(n) = n!*polcoeff( exp(x + n*x^2/2 + x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

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

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

A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2k,k)/(n-2k)!.

Original entry on oeis.org

1, 1, 1, 7, 33, 101, 1681, 14211, 72577, 1906633, 23242401, 166218911, 5966236321, 95016917997, 873707885233, 39767572858651, 781865428682241, 8787169718273681, 484500265577706817, 11335266937098816183, 150554918241183405601, 9749671976020428623221

Offset: 0

Seiichi Manyama, Apr 15 2023

nonn

Let k be a positive integer. It appears that reducing this sequence modulo k produces an eventually periodic sequence with period a multiple of k. For example, modulo 9 the sequence becomes [1, 1, 1, 7, 6, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, ...], with an apparent period [2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0] of length 18 starting at a(5). - Peter Bala, Apr 16 2023

Winston de Greef, Table of n, a(n) for n = 0..425
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362043, A362293.

Cf. A277614, A362314, A362319.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-x^3))^(1/3))/(1+lambertw(-x^3))))

a(n) = A362043(n,2*n).
a(n) = n! * [x^n] exp(x + n*x^3/3).
E.g.f.: exp( ( -LambertW(-x^3) )^(1/3) ) / (1 + LambertW(-x^3)).
a(n) ~ (1 + 2*cos(2*Pi*mod(n,3)/3 - sqrt(3)/2)/exp(3/2)) * n^n / (sqrt(3) * exp(2*n/3 - 1)). - Vaclav Kotesovec, Apr 18 2023

A362319 a(n) = n! * Sum_{k=0..floor(n/5)} (n/5)^k / (k! * (n-5*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 865, 3529, 10753, 27217, 7318081, 96720625, 689990401, 3508289929, 14239793569, 5933573525881, 114415115802625, 1165402803391009, 8298505279241857, 46355961619888993, 26167218073714552321, 663290722580370585625

Offset: 0

Seiichi Manyama, Apr 16 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..434
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277614, A362300, A362314.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-x^5))^(1/5))/(1+lambertw(-x^5))))

a(n) = n! * [x^n] exp(x + n*x^5/5).

E.g.f.: exp( ( -LambertW(-x^5) )^(1/5) ) / (1 + LambertW(-x^5)).

A362317 a(n) = n! * Sum_{k=0..floor(n/4)} (n/24)^k /(k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 5, 26, 91, 246, 2801, 26650, 159601, 702406, 12479941, 172561676, 1462655195, 8918930476, 215370384321, 3906667179836, 42828875064001, 333816101642140, 10190496077676901, 228789539391769336, 3077152545301687931, 29203537040556576776

Offset: 0

Seiichi Manyama, Apr 16 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..465
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362173, A362336.

Cf. A351932, A362314.

nmax = 30; CoefficientList[PowerExpand[Series[E^((-6*LambertW[-x^4/6])^(1/4)) / (1 + LambertW[-x^4/6]), {x, 0, nmax}]], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 18 2023 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-6*lambertw(-x^4/6))^(1/4))/(1+lambertw(-x^4/6))))

a(n) = n! * [x^n] exp(x + n*x^4/24).

E.g.f.: exp( ( -6*LambertW(-x^4/6) )^(1/4) ) / (1 + LambertW(-x^4/6)).

cp's OEIS Frontend

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

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A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2*k,k)/(n-2*k)!.

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A362319 a(n) = n! * Sum_{k=0..floor(n/5)} (n/5)^k / (k! * (n-5*k)!).

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A362317 a(n) = n! * Sum_{k=0..floor(n/4)} (n/24)^k /(k! * (n-4*k)!).

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A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2k,k)/(n-2k)!.