A115327 -id:A115327 - 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.

A115331 E.g.f.: exp(x+5/2*x^2).

Original entry on oeis.org

1, 1, 6, 16, 106, 426, 3076, 15856, 123516, 757756, 6315976, 44203776, 391582456, 3043809016, 28496668656, 241563299776, 2378813448976, 21703877431056, 223903020594016, 2177251989389056, 23448038945820576, 241173237884726176

Offset: 0

Paul D. Hanna, Jan 20 2006

nonn

Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.

a(n) is also the number of square roots of any permutation in S_{5n} whose disjoint cycle decomposition consists of n cycles of length 5. - Luis Manuel Rivera Martínez, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations , Australas. J. Combin. 52 (2012), 41-54 (Theorems 1 and 2).

Column k=5 of A359762.

Cf. A115332.

Range[0, 20]! CoefficientList[Series[Exp[(x + 5 / 2 x^2)], {x, 0, 20}], x] (* Vincenzo Librandi, May 22 2013 *)

a(n)=local(m=5);n!*polcoeff(exp(x+m/2*x^2+x*O(x^n)),n)

Term-by-term square equals A115332 which has e.g.f.: exp(x/(1-5*x))/sqrt(1-25*x^2).
a(n) ~ exp(sqrt(n/5)-n/2-1/20)*5^(n/2)*n^(n/2)/sqrt(2). - Vaclav Kotesovec, Oct 19 2012
a(n) = n!*Sum_{k=0..floor(n/2)}5^k/(2^k*k!*(n-2*k)!). - Luis Manuel Rivera Martínez, Feb 26 2015
O.g.f.: 1/(1-x - 5*x^2/(1-x - 10*x^2/(1-x - 15*x^2/(1-x - 20*x^2/(1-x - 25*x^2/(1-x -...)))))), a continued fraction (after Paul Barry in A115327). - Paul D. Hanna, Mar 08 2015

A362278 Expansion of e.g.f. exp(x - 3*x^2/2).

Original entry on oeis.org

1, 1, -2, -8, 10, 106, -44, -1952, -1028, 45820, 73576, -1301024, -3729032, 43107832, 188540080, -1621988864, -10106292464, 67749173008, 583170088672, -3075285253760, -36315980308064, 148201134917536, 2436107894325568, -7345167010231808

Offset: 0

Seiichi Manyama, Apr 13 2023

Winston de Greef, Table of n, a(n) for n = 0..690

Column k=3 of A362277.

Cf. A115327.

With[{nn=30},CoefficientList[Series[Exp[x-3 x^2/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 18 2023 *)

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

a(n) = a(n-1) - 3*(n-1)*a(n-2) for n > 1.

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

A115328 E.g.f: exp(x/(1-3x))/sqrt(1-9x^2).

Original entry on oeis.org

1, 1, 16, 100, 2116, 27556, 732736, 14776336, 476112400, 13013333776, 494512742656, 17019717246016, 747017670477376, 30923039616270400, 1542024562112889856, 74433082892402872576, 4161241771884669788416

Offset: 0

Paul D. Hanna, Jan 20 2006

nonn

Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.

Cf. A115327.

CoefficientList[Series[E^(x/(1-3*x))/Sqrt[1-9*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

a(n)=local(m=3);n!*polcoeff(exp(x/(1-m*x+x*O(x^n)))/sqrt(1-m^2*x^2+x*O(x^n)),n)

Equals term-by-term square of A115327 which has e.g.f.: exp(x+3/2*x^2).
D-finite with recurrence: a(n) = (3*n-2)*a(n-1) - 27*(n-1)*(n-2)^2*a(n-3) + 3*(n-1)*(3*n-2)*a(n-2). - Vaclav Kotesovec, Jun 26 2013
a(n) ~ 1/2*exp(-1/6+2*sqrt(n/3)-n)*3^n*n^n. - Vaclav Kotesovec, Jun 26 2013

A266503 Number of symmetric difference-closed 4-sets consisting of sets consisting of an even number of pairwise disjoint 2-subsets of {1,2,...,n}.

Original entry on oeis.org

0, 0, 0, 0, 0, 15, 105, 735, 4095, 26775, 162855, 1105335, 7187895, 51126075, 356831475, 2676468795, 19890931515, 156769986555, 1232704469115, 10178240218875, 84190426730235, 725667326178795

Offset: 1

John M. Campbell, Jan 24 2016

A set of this form forms a group (isomorphic to the Klein four-group) under the symmetric difference operation. Such sets may be regarded in a natural way as Klein four-subgroups of the alternating group A_n.

			For example, there are a(n) = 15 sets of this form in the case whereby n=6:
{{{12}, {34}}, {{34}, {56}}, {{12}, {56}}, {}}
{{{12}, {35}}, {{35}, {46}}, {{12}, {46}}, {}}
{{{12}, {36}}, {{36}, {45}}, {{12}, {45}}, {}}
{{{13}, {24}}, {{24}, {56}}, {{13}, {56}}, {}}
{{{13}, {25}}, {{25}, {46}}, {{13}, {46}}, {}}
{{{13}, {26}}, {{26}, {45}}, {{13}, {45}}, {}}
{{{14}, {23}}, {{23}, {56}}, {{14}, {56}}, {}}
{{{14}, {25}}, {{25}, {36}}, {{14}, {36}}, {}}
{{{14}, {26}}, {{26}, {35}}, {{14}, {35}}, {}}
{{{15}, {23}}, {{23}, {46}}, {{15}, {46}}, {}}
{{{15}, {24}}, {{24}, {36}}, {{15}, {36}}, {}}
{{{15}, {26}}, {{26}, {34}}, {{15}, {34}}, {}}
{{{16}, {23}}, {{23}, {45}}, {{16}, {45}}, {}}
{{{16}, {24}}, {{24}, {35}}, {{16}, {35}}, {}}
{{{16}, {25}}, {{25}, {34}}, {{16}, {34}}, {}}

John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.

Cf. A267840.

a[n_] := n!*Sum[Sum[Sum[(2^(k-2*i-2*j))/(k!*(2*i-k)!*(2*j-k)!*(n-4*i-4*j+2*k)!*(KroneckerDelta[i, j]+KroneckerDelta[i, k]+1)!), {k, Max[i, 2*i+2*j-Floor[n/2]], Min[2*j, Floor[1/4*(4*i+4*j-1)]]}], {j, 1, i}], {i, 1, Floor[n/2]}] ; Print[Table[a[n], {n, 1, 22}]] ;
Rest[CoefficientList[Series[E^x/3 - E^(-x*(x-2)/2)/8 - E^(x*(x+2)/2)/4 + E^(x*(3*x+2)/2)/24, {x, 0, 30}], x] * Range[0, 30]!] (* Vaclav Kotesovec, Apr 10 2016 *)

a(n) = n!*sum(sum(sum((2^(k-2*i-2*j))/(k!*(2*i-k)!*(2*j-k)!*(n-4*i-4*j+2*k)!*(delta(i, j)+delta(i, k)+1)!), k=max(i, 2*i+2*j-[n/2])..min(2*j, [1/4*(4*i+4*j-1)])), j=1..i), i=1..[n/2]).
From Vaclav Kotesovec, Apr 10 2016: (Start)
Recurrence: (n-6)*(n-4)*(n-2)*a(n) = (2*n - 7)*(2*n^2 - 14*n + 15)*a(n-1) + 3*(n-7)*(n-1)*(n^2 - 7*n + 11)*a(n-2) - (n-2)*(n-1)*(9*n^2 - 85*n + 189)*a(n-3) + (n-3)*(n-2)*(n-1)*(n^2 - n - 22)*a(n-4) - 2*(n-4)^2*(n-3)*(n-2)*(n-1)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n - 19)*a(n-6) + 3*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
E.g.f.: exp(x)/3 - exp(-x*(x-2)/2)/8 - exp(x*(x+2)/2)/4 + exp(x*(3*x+2)/2)/24.
a(n) ~ 2^(-7/2) * 3^(n/2 - 1) * exp(sqrt(n/3) - n/2 - 1/12) * n^(n/2).
(End)
a(n) = 1/3 + (-1)^n*A001464(n)/8 - A000085(n)/4 + A115327(n)/24. - Vaclav Kotesovec, May 28 2016

A267840 Number of symmetric difference-closed 4-sets consisting of the empty set and sets consisting of pairwise disjoint 2-subsets of {1,2,...,n}.

Original entry on oeis.org

0, 0, 0, 3, 15, 105, 525, 3255, 17703, 112455, 669735, 4485195, 29023995, 205768563, 1432735395, 10728177915, 79665069435, 627587657595, 4933313794683, 40724759240235, 336819780949995, 2902978545030795, 25135723046974155, 225455477000793963

Offset: 1

John M. Campbell, Jan 22 2016

Suppose that n people arrive at a meeting, and suppose that the n people arrange themselves into couples, except for a loner if n is odd. The couples then form various organizations. Then a(n) is the number of possible collections of three distinct organizations such that given two organizations in this collection, if a couple belongs to only one of these two organizations, then this couple must be a member of the remaining (third) organization.

A set of this form forms a group (isomorphic to the Klein four-group) under the symmetric difference operation.

			There are a(n) = 15 sets of this form in the case whereby n=5:
{{}, {{1, 2}}, {{3, 4}}, {{1, 2}, {3, 4}}}
{{}, {{1, 2}}, {{3, 5}}, {{1, 2}, {3, 5}}}
{{}, {{1, 2}}, {{4, 5}}, {{1, 2}, {4, 5}}}
{{}, {{1, 3}}, {{2, 4}}, {{1, 3}, {2, 4}}}
{{}, {{1, 3}}, {{2, 5}}, {{1, 3}, {2, 5}}}
{{}, {{1, 3}}, {{4, 5}}, {{1, 3}, {4, 5}}}
{{}, {{1, 4}}, {{2, 3}}, {{1, 4}, {2, 3}}}
{{}, {{1, 4}}, {{2, 5}}, {{1, 4}, {2, 5}}}
{{}, {{1, 4}}, {{3, 5}}, {{1, 4}, {3, 5}}}
{{}, {{1, 5}}, {{2, 3}}, {{1, 5}, {2, 3}}}
{{}, {{1, 5}}, {{2, 4}}, {{1, 5}, {2, 4}}}
{{}, {{1, 5}}, {{3, 4}}, {{1, 5}, {3, 4}}}
{{}, {{2, 3}}, {{4, 5}}, {{2, 3}, {4, 5}}}
{{}, {{2, 4}}, {{3, 5}}, {{2, 4}, {3, 5}}}
{{}, {{2, 5}}, {{3, 4}}, {{2, 5}, {3, 4}}}

John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.

Cf. A266503.

Table[n!*Sum[Sum[Sum[(2^(k-i-j))/(k!)/(i-k)!/(j-k)!/(n-2i-2j+2k)!/(KroneckerDelta[i, j]+KroneckerDelta[i, 2 k]+1)!, {k, Max[Ceiling[i/2], i+j-Floor[n/2]], Min[j, Floor[1/4*(2i+2j-1)]]}], {j, 1, i}], {i, 1, Floor[n/2]}], {n, 1, 26}]
Rest[CoefficientList[Series[E^x/3 - E^(x*(x+2)/2)/2 + E^(x*(3*x+2)/2)/6, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Apr 10 2016 *)

a(n) = n!*sum(sum(sum((2^(k-i-j))/(k!)/(i-k)!/(j-k)!/(n-2i-2j+2k)!/(delta(i, j)+delta(i, 2k)+1)!, k=max(ceiling(i/2), i+j-[n/2])..min(j, [1/4*(2i+2j-1)])), j=1..i), i=1..[n/2]), where delta denotes the Kronecker delta function.
Conjectures from Vaclav Kotesovec, Apr 10 2016: (Start)
Recurrence: (n-4)*(n-2)*a(n) = 3*(n^2 - 5*n + 5)*a(n-1) + (n-1)*(4*n^2 - 27*n + 41)*a(n-2) - (n-2)*(n-1)*(8*n - 29)*a(n-3) - (n-3)*(n-2)*(n-1)*(3*n - 16)*a(n-4) + 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
E.g.f.: exp(x)/3 - exp(x*(x+2)/2)/2 + exp(x*(3*x+2)/2)/6.
a(n) ~ 2^(-3/2) * 3^(n/2 - 1) * exp(sqrt(n/3) - n/2 - 1/12) * n^(n/2).
(End)
a(n) = 1/3 - A000085(n)/2 + A115327(n)/6. - Vaclav Kotesovec, May 28 2016

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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A115331 E.g.f.: exp(x+5/2*x^2).

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A362278 Expansion of e.g.f. exp(x - 3*x^2/2).

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A115328 E.g.f: exp(x/(1-3*x))/sqrt(1-9*x^2).

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A266503 Number of symmetric difference-closed 4-sets consisting of sets consisting of an even number of pairwise disjoint 2-subsets of {1,2,...,n}.

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A267840 Number of symmetric difference-closed 4-sets consisting of the empty set and sets consisting of pairwise disjoint 2-subsets of {1,2,...,n}.

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Mathematica

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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').

A115328 E.g.f: exp(x/(1-3x))/sqrt(1-9x^2).