A193638 -id:A193638 - cp's OEIS frontend

A114938 Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.

1, 0, 2, 30, 864, 39480, 2631600, 241133760, 29083420800, 4467125013120, 851371260364800, 197158144895712000, 54528028997584665600, 17752366094818747392000, 6720318485119046923315200, 2927066537906697348594432000, 1453437879238150456164433920000

Offset: 0

Hugo Pfoertner, Jan 08 2006

a(n) is also the number of (0,1)-matrices A=(a_ij) of size n X 2n such that each row has exactly two 1's and each column has exactly one 1 and with the restriction that no 1 stands on the line from a_11 to a_22. - Shanzhen Gao, Feb 24 2010
a(n) is the number of permutations of the multiset {1,1,2,2,...,n,n} with no fixed points. - Alexander Burstein, May 16 2020
Also the number of 2-uniform ordered set partitions of {1...2n} containing no two successive vertices in the same block. - Gus Wiseman, Jul 04 2020

			a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.

R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.

Seiichi Manyama, Table of n, a(n) for n = 0..238 (terms 1..100 from Andrew Woods)
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.

Cf. A114939 = preferred seating arrangements of n couples.
Cf. A007060 = arrangements of n couples with no adjacent spouses; A007060(n) = 2^n * A114938(n) (this sequence).
Cf. A104548, A193638.
Cf. A278990 = number of loopless linear chord diagrams with n chords.
Cf. A000806 = Bessel polynomial y_n(-1).
The version for multisets with prescribed multiplicities is A335125.
The version for prime indices is A335452.
Anti-run compositions are counted by A003242.
Anti-run compositions are ranked by A333489.
Inseparable partitions are counted by A325535.
Inseparable partitions are ranked by A335448.
Separable partitions are counted by A325534.
Separable partitions are ranked by A335433.
Cf. A007716, A128695, A292884, A335126, A335434, A335451.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.
Row n=2 of A322093.

[1] cat [n le 2 select 2*(n-1) else n*(2*n-1)*Self(n-1) + (n-1)*n*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Aug 10 2015

Table[Sum[Binomial[n,i](2n-i)!/2^(n-i) (-1)^i,{i,0,n}],{n,0,20}]  (* Geoffrey Critzer, Jan 02 2013, and adapted to the extension by Stefano Spezia, Nov 15 2018 *)
Table[Length[Select[Permutations[Join[Range[n],Range[n]]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,5}] (* Gus Wiseman, Jul 04 2020 *)
A114938[n_] := ((2 n)! Hypergeometric1F1[-n, -2 n, -2]) / 2^n;
Array[A114938, 17, 0]  (* Peter Luschny, Sep 04 2025 *)

A114938(n)=sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!/2^k);
vector(20, n, A114938(n-1)) \\ Michel Marcus, Aug 10 2015

def A114938(n): return (-1)^n*sum(binomial(n,k)*factorial(n+k)//(-2)^k for k in range(n+1))
[A114938(n) for n in range(31)] # G. C. Greubel, Sep 26 2023

a(n) = Sum_{k=0..n} ((binomial(n, k)*(-1)^(n-k)*(n+k)!)/2^k).
a(n) = (-1)^n * n! * A000806(n), n>0. - Vladeta Jovovic, Nov 19 2009
a(n) = n*(2*n-1)*a(n-1) + (n-1)*n*a(n-2). - Vaclav Kotesovec, Aug 07 2013
a(n) ~ 2^(n+1)*n^(2*n)*sqrt(Pi*n)/exp(2*n+1). - Vaclav Kotesovec, Aug 07 2013
a(n) = n! * A278990(n). - Alexander Burstein, May 16 2020
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*sqrt(2/Pi) * n! * BesselK(n+1/2, -1).
a(n) = [n! * (1/x) * p_{n+1}(x)]|A104548%20for%20p">{x= -1} (See A104548 for p{n}(x)).
E.g.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * erfi((1+x)/sqrt(2*x)).
Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(sqrt(1-2*x))/sqrt(1-2*x).
Sum_{n >= 0} a(n)*x^n/(n!*(n+1)!) = ( 1 - exp(-1 + sqrt(1-2*x)) )/x. (End)
a(n) = ((2*n)!/2^n) * hypergeom([-n], [-2*n], -2]) = A007060(n) / 2^n. - Peter Luschny, Sep 04 2025

a(0)=1 prepended by Seiichi Manyama, Nov 15 2018

A322093 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 2, 0, 6, 2, 0, 24, 30, 2, 0, 120, 864, 174, 2, 0, 720, 39480, 41304, 1092, 2, 0, 5040, 2631600, 19606320, 2265024, 7188, 2, 0, 40320, 241133760, 16438575600, 11804626080, 134631576, 48852, 2, 0, 362880, 29083420800, 22278418248240, 131402141197200, 7946203275000, 8437796016, 339720, 2, 0

Offset: 1

Seiichi Manyama, Nov 26 2018

			Square array begins:
   1, 2,    6,        24,           120,                 720, ...
   0, 2,   30,       864,         39480,             2631600, ...
   0, 2,  174,     41304,      19606320,         16438575600, ...
   0, 2, 1092,   2265024,   11804626080,     131402141197200, ...
   0, 2, 7188, 134631576, 7946203275000, 1210527140790855600, ...

Seiichi Manyama, Antidiagonals n = 1..52, flattened
Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.

Columns k=3 gives A110706.
Rows n=1..10 give A000142, A114938, A193638, A321633, A322126, A321382, A322095, A322096, A322145, A322146.
Main diagonal gives A321634.
Cf. A322013.

Table[Table[SeriesCoefficient[1/(1 - Sum[x[i]/(1 + x[i]), {i, 1, n}]), Sequence @@ Table[{x[i], 0, k}, {i, 1, n}]],{n, 1, 6}], {k, 1, 5}] (* Zlatko Damijanic, Nov 03 2024 *)

q(n,x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n,k) = subst(serlaplace(q(n,x)^k), x, 1) \\ Andrew Howroyd, Feb 03 2024

A(n,k) = k! * A322013(n,k).
Let q_n(x) = Sum_{i=1..n} (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!.
A(n,k) = Integral_{0..infinity} (q_n(x))^k * exp(-x) dx.

A190826 Number of permutations of 3 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 29, 1721, 163386, 22831355, 4420321081, 1133879136649, 372419001449076, 152466248712342181, 76134462292157828285, 45552714996556390334921, 32173493282909179882613934, 26487410329744429030530295991, 25143126122564855343240882599761, 27260957330891104469298062949026065

Offset: 0

R. H. Hardin, May 21 2011

nonn

			Some of the a(3) = 29 solutions for n=3: 123232131, 123121323, 123123213, 123212313, 123213123, 121323132, 123132312, 123123123, 123231213, 121323123, 121321323, 121312323, 121323231, 123231321, 121313232, 123132321, ...

Seiichi Manyama, Table of n, a(n) for n = 0..223 (terms 0..101 from Andrew Woods)
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{c,3}/3!.

Cf. A192990, A193624, A193638.

Row n=3 of A322013.

B:=Binomial;
f:= func< n,j | (&+[B(n,k)*B(2*k,j)*(-3)^(k-j): k in [Ceiling(j/2)..n]]) >;
A190826:= func< n | (-1/2)^n*(&+[Factorial(j)*B(n+j,j)*f(n,j): j in [0..2*n]]) >;
[A190826(n): n in [0..30]]; // G. C. Greubel, Sep 22 2023

a[n_]:= 1/(6^n*n!)*Sum[(n+j)! Sum[Binomial[n,k] Binomial[2k,j] (-3)^(n+k-j), {k, Ceiling[j/2], n}], {j,0,2n}]; Array[a, 16, 0] (* Jean-François Alcover, Jul 22 2017, after Tani Akinari's code for A193638 *)

b=binomial;
def f(j,n): return sum(b(n,k)*b(2*k,j)*(-3)^(k-j) for k in range((j//2),n+1))
def A190826(n): return (-1/2)^n*sum(factorial(j)*b(n+j,j)*f(j,n) for j in range(2*n+1))
[A190826(n) for n in range(31)] # G. C. Greubel, Sep 22 2023

a(n) = A193624(n)/(6^n * n!), for n >= 1.
a(n) = A193638(n)/n!, for n >= 1.
a(n) = A192990(binomial(n+2,3)) / (6^n * n!), for n >= 1.
2*a(n) -3*(3*n^2-3*n+4)*a(n-1) +2*(9*n^2-42*n+47)*a(n-2) +8*(3*n-7)*a(n-3) -8*a(n-4) = 0. - R. J. Mathar, May 23 2014
a(n) = (1/(6^n * n!)) * Sum_{j=0..2*n} Sum_{k=ceiling(j/2)..n} (n+j)! * binomial(2*k, j) * binomial(n, k) * (-3)^(n+k-j). - Jean-François Alcover, Jul 22 2017
a(n) ~ 3^(2*n + 1/2) * n^(2*n) / (2^n * exp(2*n + 2)). - Vaclav Kotesovec, Nov 24 2018

a(0)=1 prepended by Alois P. Heinz, Jul 22 2017

A321633 Number of permutations of the multiset {1,1,1,1,2,2,2,2,3,3,3,3,...,n,n,n,n} with no two consecutive terms equal.

Original entry on oeis.org

1, 0, 2, 1092, 2265024, 11804626080, 131402141197200, 2778291737177034960, 102284730928300590754560, 6134232798447803932455457920, 568598490353320413296928514444800, 78076149156802562231395694989534464000, 15336188146163145199585928509793662920345600

Offset: 0

Seiichi Manyama, Nov 15 2018

nonn

			a(2) = 2 because there are two permutations of {1,1,1,1,2,2,2,2} avoiding equal consecutive terms: 12121212 and 21212121.

Seiichi Manyama, Table of n, a(n) for n = 0..129
Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.

Row 4 of A322093.

Cf. A000142, A114938, A190830, A193638.

a[n_] := Integrate[(-x + 3/2 * x^2 - 1/2 * x^3 + 1/24 * x^4)^n  * Exp[-x],  {x, 0, Infinity}]; Array[a, 10, 0] (* Stefano Spezia, Nov 27 2018 *)

a(n) = n! * A190830(n).

a(n) = Integral_{0..oo} (-x + 3/2 * x^2 - 1/2 * x^3 + 1/24 * x^4)^n * exp(-x) dx.

A321634 Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms.

Original entry on oeis.org

1, 1, 2, 174, 2265024, 7946203275000, 12229789732207993835280, 12202002913678756821228939869239920, 10937192762438008527903830198163831816546577931520, 11655577382287102750765311537460065620507094071664576111302628243840

Offset: 0

Seiichi Manyama, Nov 15 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26
Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.

Cf. A000142, A110706, A114938, A193638, A321633, A321666.

{a(n) = sum(i=n, n^2, i!*polcoef(sum(j=1, n, (-1)^(n-j)*binomial(n-1, j-1)*x^j/j!)^n, i))} \\ Seiichi Manyama, May 27 2019

a(n) ~ n^(n^2 - n/2 + 1) / ((2*Pi)^((n-1)/2) * exp(n - 5/12)). - Vaclav Kotesovec, Nov 24 2018

cp's OEIS Frontend

A114938 Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.

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A322093 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k with no element equal to another within a distance of 1.

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A190826 Number of permutations of 3 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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A321633 Number of permutations of the multiset {1,1,1,1,2,2,2,2,3,3,3,3,...,n,n,n,n} with no two consecutive terms equal.

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A321634 Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms.

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