A277614 -id:A277614 - cp's OEIS frontend

A359762 Array read by ascending antidiagonals. T(n, k) = n![x^n] exp(x + (k/2) x^2). A generalization of the number of involutions (or of 'telephone numbers').

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 10, 7, 4, 1, 1, 1, 26, 25, 10, 5, 1, 1, 1, 76, 81, 46, 13, 6, 1, 1, 1, 232, 331, 166, 73, 16, 7, 1, 1, 1, 764, 1303, 856, 281, 106, 19, 8, 1, 1, 1, 2620, 5937, 3844, 1741, 426, 145, 22, 9, 1, 1

Offset: 0

Peter Luschny, Jan 14 2023

The array is a generalization of the number of involutions of permutations on n letters, A000085, also known as 'telephone numbers'. According to Bednarz et al. the telephone number interpretation "is due to John Riordan, who noticed that T(n, 1) is the number of connection patterns in a telephone system with n subscribers."

In graph theory, the n-th telephone number is the total number of matchings of a complete graph K_n (see the Wikipedia entry). Assuming a network with k possibilities of connections leads to a network that can be modeled by a complete multigraph K(n, k). The total number of connection patterns in such a network is given by T(n, k).

			Array T(n, k) starts:
  [n\k] 0   1      2        3       4        5        6        7
  --------------------------------------------------------------
  [0] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
  [1] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
  [2] 1,    2,     3,       4,      5,       6,       7,       8, ... [A000027]
  [3] 1,    4,     7,      10,     13,      16,      19,      22, ... [A016777]
  [4] 1,   10,    25,      46,     73,     106,     145,     190, ... [A100536]
  [5] 1,   26,    81,     166,    281,     426,     601,     806, ...
  [6] 1,   76,   331,     856,   1741,    3076,    4951,    7456, ...
  [7] 1,  232,  1303,    3844,   8485,   15856,   26587,   41308, ...
  [8] 1,  764,  5937,   21820,  57233,  123516,  234529,  406652, ...
  [9] 1, 2620, 26785,  114076, 328753,  757756, 1510705, 2719900, ...
   [A000085][A047974][A115327][A115329][A115331]

John Riordan, Introduction to Combinatorial Analysis, Dover (2002).

Urszula Bednarz and Małgorzata Wołowiec-Musiał, On a new generalization of telephone numbers, Turkish Journal of Mathematics: Vol. 43: No. 3, (2019).
Carlos M. da Fonseca and Anthony G. Shannon, Telephone numbers extensions, J. Interdisc. Math. (2025) Vol. 28, No. 4, 1573-1580. See pp. 1574, 1577.
Wikipedia, Telephone numbers.

Rows include: A000012, A000027, A016777, A100536.
Columns include: A000012, A000085, A047974, A115327, A115329, A115331, A293720.
Cf. A000085, A277614, A359760.

T := (n, k) -> add(binomial(n, j)*doublefactorial(j-1)*k^(j/2), j = 0..n, 2):
for n from 0 to 9 do lprint(seq(T(n, k), k = 0..7)) od;
T := (n, k) -> ifelse(k=0, 1, I^(-n)*(2*k)^(n/2)*KummerU(-n/2, 1/2, -1/(2*k))):
seq(seq(simplify(T(n-k, k)), k = 0..n), n = 0..10);
T := proc(n, k) exp(x + (k/2)*x^2): series(%, x, 16): n!*coeff(%, x, n) end:
seq(lprint(seq(simplify(T(n, k)), k = 0..8)), n = 0..9);
T := proc(n, k) option remember; if n = 0 or n = 1 then 1 else T(n, k-1) +
n*(k-1)*T(n, k-2) fi end: for n from 0 to 9 do seq(T(n, k), k=0..9) od;
# Only to check the interpretation as a determinant of a lower Hessenberg matrix:
gen := proc(i, j, n) local ev, tv; ev := irem(j+i, 2) = 0; tv := j < i and not ev;
if j > i + 1 then 0 elif j = i + 1 then -1 elif j <= i and ev then 1
elif tv and i < n then x*(n + 1 - i) - 1 else x fi end:
det := M -> LinearAlgebra:-Determinant(M):
p := (n, k) -> subs(x = k, det(Matrix(n, (i, j) -> gen(i, j, n)))):
for n from 0 to 9 do seq(p(n, k), k = 0..7) od;

T[n_, k_] := Sum[Binomial[n, j] Factorial2[j-1] * If[j==0, 1,  k^(j/2)], {j, 0, n, 2}];
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from math import factorial, comb
def oddfactorial(n: int) -> int:
    return factorial(2 * n) // (2**n * factorial(n))
def T(n: int, k: int) -> int:
    return sum(comb(n, 2 * j) * oddfactorial(j) * k**j for j in range(n + 1))
for n in range(10): print([T(n, k) for k in range(8)])

T(n, k) = Sum_{j=0..n, j even} binomial(n, j) * (j - 1)!! * k^(j/2).
T(n, k) = T(n, k-1) + n*(k-1)*T(n, k-2) for n >= 2, T(n, 0) = T(n, 1) = 1.
T(n, k) = i^(-n) * (2*k)^(n/2) * KummerU(-n/2, 1/2, -1/(2*k)) for k >= 1, and T(n, 0) = 1.

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

A362276 a(n) = n! * Sum_{k=0..floor(n/2)} (-n/2)^k * binomial(n-k,k)/(n-k)!.

Original entry on oeis.org

1, 1, -1, -8, 25, 326, -1709, -31016, 228257, 5311900, -50337449, -1429574464, 16573668409, 555724876552, -7619288730325, -294582728145824, 4662562423032961, 204200579987319824, -3664348770051277073, -179294278761195862400, 3597007651803106610201

Offset: 0

Seiichi Manyama, Apr 13 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Main diagonal of A362277.

Cf. A277614.

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

a(n) = n! * [x^n] exp(x - n*x^2/2).

E.g.f.: exp( sqrt( LambertW(x^2) ) ) / (1 + LambertW(x^2)).

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)).

A362347 a(n) = n! * Sum_{k=0..floor(n/2)} k^k / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 3, 7, 61, 261, 3991, 24403, 524217, 4149001, 114544171, 1111976031, 37492210933, 431097055117, 17165526306111, 228085258466731, 10472666396599921, 157882659583461393, 8211536252680154707, 138474928851961700791, 8045878340298511456941

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A086331, A362348, A362349.

Cf. A277614.

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

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

a(n) ~ (exp(2*exp(-1/2)) + (-1)^n) * n^n / (sqrt(2) * exp(n/2 + exp(-1/2))). - Vaclav Kotesovec, Aug 05 2025

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)

A362350 a(n) = n! * Sum_{k=0..floor(n/2)} (k/2)^k / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 2, 4, 19, 71, 601, 3277, 39089, 277489, 4250341, 37110701, 693581197, 7184750509, 158461520309, 1899055549861, 48269252293201, 656869268651537, 18903165795857089, 287927838327392929, 9252988524143245181, 155954097639111859501

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..437
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362351, A362352.

Cf. A277614.

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

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

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

A362281 a(n) = n! * Sum_{k=0..floor(n/2)} n^k * binomial(n-k,k)/(n-k)!.

Original entry on oeis.org

1, 1, 5, 19, 241, 1601, 32581, 308995, 8655809, 106673761, 3805452901, 57704760851, 2500580809585, 45018720191329, 2295683481085541, 47848514992963651, 2806491306922172161, 66464103165835330625, 4407449313521981148229, 116893033842508769526931

Offset: 0

Seiichi Manyama, Apr 14 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..395
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277614, A362282.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(-lambertw(-2*x^2)/2))/(1+lambertw(-2*x^2))))

a(n) = n! * [x^n] exp(x + n*x^2).
E.g.f.: exp( sqrt( -LambertW(-2*x^2)/2 ) ) / (1 + LambertW(-2*x^2)).
a(n) ~ (1 + (-1)^n/exp(sqrt(2))) * 2^((n-1)/2) * n^n / exp(n/2 - 1/sqrt(2)). - Vaclav Kotesovec, Apr 15 2023

A362283 Expansion of e.g.f. exp( sqrt(-LambertW(-x^2)) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 106, 601, 7456, 60649, 1012348, 10748161, 225641296, 2957978101, 74847384184, 1168123938073, 34598428916416, 626497273410961, 21261683280971536, 438222313050326209, 16765636110497697088, 387549609831150094621, 16502188154766906299296

Offset: 0

Seiichi Manyama, Apr 14 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..417
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000272, A034940, A277614, A362293.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(-lambertw(-x^2)))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} A034940(k) * binomial(n-1,2*k) * a(n-2*k-1).

cp's OEIS Frontend

A359762 Array read by ascending antidiagonals. T(n, k) = n!*[x^n] exp(x + (k/2) * x^2). A generalization of the number of involutions (or of 'telephone numbers').

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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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A362276 a(n) = n! * Sum_{k=0..floor(n/2)} (-n/2)^k * binomial(n-k,k)/(n-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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A362347 a(n) = n! * Sum_{k=0..floor(n/2)} k^k / (k! * (n-2*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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A362350 a(n) = n! * Sum_{k=0..floor(n/2)} (k/2)^k / (k! * (n-2*k)!).

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A362281 a(n) = n! * Sum_{k=0..floor(n/2)} n^k * binomial(n-k,k)/(n-k)!.

A359762 Array read by ascending antidiagonals. T(n, k) = n![x^n] exp(x + (k/2) x^2). A generalization of the number of involutions (or of 'telephone numbers').

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

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