A362300 -id:A362300 - 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

A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2j,j)/(n-2j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 4, 9, 11, 1, 1, 1, 1, 5, 13, 21, 31, 1, 1, 1, 1, 6, 17, 31, 81, 106, 1, 1, 1, 1, 7, 21, 41, 151, 351, 337, 1, 1, 1, 1, 8, 25, 51, 241, 736, 1233, 1205, 1, 1, 1, 1, 9, 29, 61, 351, 1261, 2689, 5769, 5021, 1

Offset: 0

Seiichi Manyama, Apr 15 2023

			Square array begins:
  1,  1,  1,   1,   1,   1,   1, ...
  1,  1,  1,   1,   1,   1,   1, ...
  1,  1,  1,   1,   1,   1,   1, ...
  1,  2,  3,   4,   5,   6,   7, ...
  1,  5,  9,  13,  17,  21,  25, ...
  1, 11, 21,  31,  41,  51,  61, ...
  1, 31, 81, 151, 241, 351, 481, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000012, A190865, A001470.
Main diagonal gives A362173.
T(n,2*n) gives A362300.
T(n,6*n) gives A362301.
Cf. A359762, A362302.

T(n, k) = n!*sum(j=0, n\3, (k/6)^j/(j!*(n-3*j)!));

E.g.f. of column k: exp(x + k*x^3/6).
T(n,k) = T(n-1,k) + k * binomial(n-1,2) * T(n-3,k) for n > 2.
T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j / (j! * (n-3*j)!).

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)).

A362173 a(n) = n! * Sum_{k=0..floor(n/3)} (n/6)^k * binomial(n-2k,k)/(n-2k)!.

Original entry on oeis.org

1, 1, 1, 4, 17, 51, 481, 3676, 18369, 272917, 3011201, 21058236, 427112401, 6160655359, 55380250017, 1423658493076, 25361574327041, 278603741558601, 8673295084155649, 183914415577719892, 2387417408385462801, 87273239189497636171, 2146479566819857007201

Offset: 0

Seiichi Manyama, Apr 14 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..441
Eric Weisstein's World of Mathematics, Lambert W-Function.

Main diagonal of A362043.

Cf. A277614, A362300, A362301.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-2*lambertw(-x^3/2))^(1/3))/(1+lambertw(-x^3/2))))

a(n) = n! * [x^n] exp(x + n*x^3/6).

E.g.f.: exp( ( -2*LambertW(-x^3/2) )^(1/3) ) / (1 + LambertW(-x^3/2)).

A362314 a(n) = n! * Sum_{k=0..floor(n/4)} (n/4)^k /(k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 25, 151, 541, 1471, 84001, 925345, 5682601, 25177681, 2245355641, 35901100951, 312222474565, 1917363070351, 232479594721921, 4873115730725761, 54830346428307601, 430468886732009185, 65997947903313461401, 1711564302775814535511

Offset: 0

Seiichi Manyama, Apr 15 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..431
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277614, A362300, A362319.

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

a(n) = n! * [x^n] exp(x + n*x^4/4).
E.g.f.: exp( ( -LambertW(-x^4) )^(1/4) ) / (1 + LambertW(-x^4)).
From Vaclav Kotesovec, Apr 18 2023: (Start)
a(n) ~ c * n^n / exp(3*n/4), where
c = cosh(1) + cos(1) if mod(n,4)=0,
c = sinh(1) + sin(1) if mod(n,4)=1,
c = cosh(1) - cos(1) if mod(n,4)=2,
c = sinh(1) - sin(1) if mod(n,4)=3. (End)

A362304 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, -5, -31, -99, 1201, 13231, 70785, -1362311, -21562399, -161746749, 4263108961, 87979472725, 849097038609, -28416142768649, -723086288422399, -8532476619366159, 346207723221680065, 10474480743776327179, 146105160034616914401

Offset: 0

Seiichi Manyama, Apr 15 2023

sign

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

Cf. A362300, A362302.

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

a(n) = A362302(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)).

A362293 Expansion of e.g.f. exp( (-LambertW(-x^3))^(1/3) ).

Original entry on oeis.org

1, 1, 1, 1, 9, 41, 121, 2241, 18481, 91729, 2577681, 30833441, 215554681, 8126363961, 127462383049, 1150296157921, 54416525377761, 1056352067669921, 11684649751431841, 665061201610232769, 15390714465319910761, 201615391902487799881

Offset: 0

Seiichi Manyama, Apr 14 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 3 the sequence becomes [1, 1, 1, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, ...], with an apparent period [1, 1, 0, 2, 1, 0] of length 6 starting at a(1). - Peter Bala, Apr 16 2023

Seiichi Manyama, Table of n, a(n) for n = 0..428
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000272, A362283, A362292, A362300.

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

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} A362292(k) * binomial(n-1,3*k) * a(n-3*k-1).

a(n) ~ (1 + 2*cos(2*Pi*mod(n-1,3)/3 - sqrt(3)/2)/exp(3/2)) * n^(n-1) / (sqrt(3) * exp(2*n/3 - 1)). - Vaclav Kotesovec, Apr 18 2023

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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A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2*j,j)/(n-2*j)!.

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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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A362173 a(n) = n! * Sum_{k=0..floor(n/3)} (n/6)^k * binomial(n-2*k,k)/(n-2*k)!.

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A362314 a(n) = n! * Sum_{k=0..floor(n/4)} (n/4)^k /(k! * (n-4*k)!).

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A362304 a(n) = n! * Sum_{k=0..floor(n/3)} (-n/3)^k * binomial(n-2*k,k)/(n-2*k)!.

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A362293 Expansion of e.g.f. exp( (-LambertW(-x^3))^(1/3) ).

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A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2j,j)/(n-2j)!.

A362173 a(n) = n! * Sum_{k=0..floor(n/3)} (n/6)^k * binomial(n-2k,k)/(n-2k)!.

A362304 a(n) = n! * Sum_{k=0..floor(n/3)} (-n/3)^k * binomial(n-2k,k)/(n-2k)!.