A010790 -id:A010790 - cp's OEIS frontend

A215943 Number of ways to place k non-attacking bishops on an n x n toroidal chessboard, summed over all k >= 0.

2, 9, 34, 289, 1546, 19321, 130922, 2169729, 17572114, 364466281, 3405357682, 85143154849, 896324308634, 26309790300249, 306827170866106, 10366719612433921, 132240988644215842, 5064730099043043529, 69974827707903049154, 3000912883089564050721

Offset: 1

Vaclav Kotesovec, Aug 28 2012

nonn

a(n) = A002720(n) if n is odd.

Vaclav Kotesovec, Table of n, a(n) for n = 1..440
V. Kotesovec, Non-attacking chess pieces

Cf. A002720, A201862, A010790, A177755-A177759, A178140, A189789, A189790, A189791.

Table[Sum[If[EvenQ[n],2^k*k!*Sum[Binomial[n/2,i]^2*Binomial[n/2,k-i]^2/Binomial[k,i],{i,0,k}],Binomial[n,k]^2*k!],{k,0,n}],{n,1,25}]

Recurrence: a(n) = ((12*n^5 - 158*n^4 - (6*(-1)^n-706)*n^3 - (1193-41*(-1)^n)*n^2 - 8*(7*(-1)^n-72)*n - 22*(-1)^n-28)*a(n-2) + (-12*n^6 + 206*n^5 + 2*(7*(-1)^n-691)*n^4 + (4545-137*(-1)^n)*n^3 + (442*(-1)^n-7442)*n^2 + (5194-544*(-1)^n)*n + 198*(-1)^n-698)*(n-2)*a(n-4) + 2*(2*n-1)*(n^2-7*n+10)^2*(n-4)^4*a(n-6))/(2*(n-5)^2*(2*n-5)).

A258213 Number of permutations of {1,2,3,...,n} such that no even numbers are adjacent.

Original entry on oeis.org

1, 1, 2, 6, 12, 72, 144, 1440, 2880, 43200, 86400, 1814400, 3628800, 101606400, 203212800, 7315660800, 14631321600, 658409472000, 1316818944000, 72425041920000, 144850083840000, 9560105533440000, 19120211066880000, 1491376463216640000, 2982752926433280000

Offset: 0

Ran Pan, May 23 2015

nonn

Cf. A008279, A010790.

a:= n-> (m-> m!^2*(m+1))(iquo(n+1, 2, 'r'))/(2-r):
seq(a(n), n=0..24);  # Alois P. Heinz, Feb 14 2024

T(n,k) = n!/(n-k)!; \\ A008279
a(n) = ceil(n/2)!*T(ceil(n/2)+1, n\2); \\ Michel Marcus, Nov 24 2022

a(n) = factorial(ceiling(n/2))*fallfac(ceiling(n/2)+1, floor(n/2)), with fallfac = A008279.
a(2n) = A010790(n), a(2n-1) = A010790(n)/2.
D-finite with recurrence: (4*(n-2)^2 + 24*n - 80)*a(n) + (16*n+24)*a(n-1) - (n+2)*n*((n-2)^2 + 8*n - 17)*a(n-2) = 0. - Georg Fischer, Nov 23 2022

A369597 a(n) is the number of reduced stable marriage problem instances of order 3 that generate n possible stable matchings.

Original entry on oeis.org

2840, 957, 91

Offset: 1

Dan Eilers, Jan 27 2024

nonn

Cf. A351430 (order 4, reduced), A368419 (order 5, reduced).
Cf. A344666 (order 3 unreduced), A344667 (order 4 unreduced).
Cf. A351409 (number of reduced instances of order n).
Cf. A010790 (reduction factor for order n).

A084259 a(n) = n! * (n+1)! * (n+2)! * (n+3).

Original entry on oeis.org

6, 48, 1440, 103680, 14515200, 3483648000, 1316818944000, 737418608640000, 584035538042880000, 630758381086310400000, 901984484953423872000000, 1666867328193927315456000000, 3900469547973789918167040000000

Offset: 0

Benoit Cloitre, Jun 21 2003

nonn

Cf. A000142, A010790, A176037.

a[n_] := n! * (n+1)! * (n+2)! * (n+3); Array[a, 15, 0] (* Amiram Eldar, May 07 2025 *)

Definition adapted to data by Georg Fischer, May 10 2021

A176049 Primes of the form n!(n+1)!(n+2)! - 1 or n!(n+1)!(n+2)! + 1.

Original entry on oeis.org

3, 11, 13, 2073601, 146313215999, 52563198423859200001, 709885457731229765106401279999999, 15120395453651827088974983182763034097693491200000000001

Offset: 1

Jonathan Vos Post, Apr 07 2010

Primes of the form A010790(k)-1 or A010790(k)+1. This is the 3rd sequence in the supersequence whose first member is factorial primes, A002981 UNION A002982, and whose 2nd member is A176038 Primes of the form n!*(n+1)! - 1 or n!*(n+1)! + 1.
a(9) has already 486 digits and is not listed for that reason. The sequence is generated by the n-values 0, 1, 1, 4, 6, 9, 13, 19, 101, 196,... [From R. J. Mathar, Oct 03 2010]
a(9) also ends with 72 nines. - Harvey P. Dale, Jan 05 2013

			a(2) = 11 because 1!*(1+1)!*(1+2)! - 1 = 11 is prime. a(4) = 2073601 because 4!*(4+1)!*(4+2)! + 1 = 2073601 is prime. a(7) because 13!*(13+1)!*(13+2)! - 1 = 709885457731229765106401279999999 is prime.

Cf. A000040, A000142, A002981, A002982, A010790, A176038.

Select[Union[Flatten[Times@@#+{1,-1}&/@Partition[Range[0,30]!,3,1]]], PrimeQ] (* Harvey P. Dale, Jan 05 2013 *)

a(8) from R. J. Mathar, Oct 03 2010

A258216 Number of permutations of {1,2,3,...,n} such that no multiples of 3 are adjacent.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 480, 3600, 30240, 151200, 1693440, 20321280, 121927680, 1828915200, 28740096000, 201180672000, 3793692672000, 73977007104000, 591816056832000, 13463815292928000, 314155690168320000

Offset: 0

Ran Pan, May 23 2015

Cf. A008279, A010790, A258213.

Table[Factorial[n-Floor[n/3]]*FactorialPower[n+1-Floor[n/3],Floor[n/3],1], {n, 0, 20}]

a(n) = factorial(n-floor(n/3))*fallfac(n+1-floor(n/3),floor(n/3)), where fallfac is falling factorial, A008279.

a(n) ~ n! * 2^(2*n+2-2*floor(n,3)/3) / 3^(n+1/2). - Vaclav Kotesovec, May 23 2015

A273935 Number of ways to arrange n women and n men around a circular table so that they can be divided into n nonintersecting pairs of 1 woman and 1 man sitting side-by-side.

Original entry on oeis.org

1, 1, 6, 84, 2160, 89280, 5443200, 460857600, 51819264000, 7476605337600, 1347105779712000, 296508121620480000, 78297264318873600000, 24431729220414996480000, 8893692297263669575680000, 3735464765667589501747200000, 1793050447716486548029440000000

Offset: 0

Vincent Chan, Jun 16 2016

nonn

a(n) is the number of monotonic paths from (0,0) to (n,n) which are equivalent to a path which is no more than 1 step from the main diagonal, where two paths are equivalent if they are circular shifts of one another.

			For n = 2: Represent the women and men as the numbers 1,2,3,4, where 1,3 are women and 2,4 are men. Then all 6 arrangements around the circular table are valid (parentheses included to emphasize a valid pairing, including wraparound):
(1,2),(3,4);
(1,2),(4,3);
1),(3,2),(4;
1),(3,4),(2;
(1,4),(2,3);
(1,4),(3,2).

Cf. A000142, A000225, A010790.

a(n) = if (n==0, 1, n!*(n-1)!*(2^n-1)); \\ Michel Marcus, Jun 20 2016

a(n) = n!*(n-1)!*(2^n-1) for n>0, a(0) = 1.

a(n) = A000142(n)*A000142(n-1)*A000225(n), n >= 1. - Omar E. Pol, Jun 16 2016

A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 6, 12, 12, 1, 24, 144, 288, 288, 1, 120, 2880, 17280, 34560, 34560, 1, 720, 86400, 2073600, 12441600, 24883200, 24883200, 1, 5040, 3628800, 435456000, 10450944000, 62705664000, 125411328000, 125411328000

Offset: 0

Werner Schulte, Jul 08 2020

Based on some integer sequence a(n), n>0, define triangular arrays A(a;n,k) by recurrence: A(a;0,0) = 1, and A(a;i,j) = 0 if j<0 or j>i, and A(a;n,k) = n! / (n-k)! * A(a;n-1,k) + a(n) * A(a;n-1,k-1) for 0<=k<=n. Then, Product_{i=1..n} (1 + (a(i) / i!) * x) = Sum_{k=0..n} A(a;n,k) / T(n,k) * x^k for n>=0 with empty product 1 (case n=0).

For the row reversed triangle R(n,k) = Product_{i=k+1..n} i! with empty product 1 (case k=n) the terms of the matrix inverse M are given by M(n,n) = 1 for n >= 0 and M(n,n-1) = -n! for n > 0 otherwise 0. - Werner Schulte, Oct 25 2022

			The triangle starts:
n\k :  0     1      2        3         4         5         6
============================================================
  0 :  1
  1 :  1     1
  2 :  1     2      2
  3 :  1     6     12       12
  4 :  1    24    144      288       288
  5 :  1   120   2880    17280     34560     34560
  6 :  1   720  86400  2073600  12441600  24883200  24883200
  etc.

Cf. A000012 (col_0), A000142 (col_1), A010790 (col_2), A176037 (col_3), A000178 (main diagonal and first subdiagonal).
Row sums equal A051399(n+1).
Cf. A009963, A093002, A193520.

T[n_, k_] := Product[i!, {i, n - k + 1, n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 08 2020 *)

T(n,k) = T(n,1) * T(n-1,k-1) for 0 < k <= n.
T(2*n,n) = A093002(n+1) for n >= 0.
T(n,k)/T(k,k) = A009963(n,k) for 0 <= k <= n.
(Sum_{k=0..n} T(n,k) * T(n,n-k))/T(n,n) = A193520(n) for n >= 0.

A362588 Triangle read by rows, T(n, k) = RisingFactorial(n - k, k) * FallingFactorial(n, k).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 12, 0, 1, 12, 72, 144, 0, 1, 20, 240, 1440, 2880, 0, 1, 30, 600, 7200, 43200, 86400, 0, 1, 42, 1260, 25200, 302400, 1814400, 3628800, 0, 1, 56, 2352, 70560, 1411200, 16934400, 101606400, 203212800, 0

Offset: 0

Peter Luschny, May 05 2023

			Table T(n, k) begins:
[0] 1;
[1] 1,  0;
[2] 1,  2,    0;
[3] 1,  6,   12,     0;
[4] 1, 12,   72,   144,       0;
[5] 1, 20,  240,  1440,    2880,        0;
[6] 1, 30,  600,  7200,   43200,    86400,         0;
[7] 1, 42, 1260, 25200,  302400,  1814400,   3628800,         0;
[8] 1, 56, 2352, 70560, 1411200, 16934400, 101606400, 203212800, 0;

Cf. A228229 (row sums), A002378 (column 1), A010790 (subdiagonal).

T := (n, k) -> (-1)^k*pochhammer(n - k, k)*pochhammer(-n, k):
for n from 0 to 6 do seq(T(n, k), k=0..n) od;

T(n, k) = (-1)^k * Pochhammer(n - k, k) * Pochhammer(-n, k).

T(n, k) = binomial(n, k) * binomial(n - 1, k) * (k!)^2.

A368787 a(n) = (n+1) * (n!)^2 * Sum_{k=1..n} 1/((k+1) * (k!)^2).

Original entry on oeis.org

0, 1, 7, 85, 1701, 51031, 2143303, 120024969, 8641797769, 777761799211, 85553797913211, 11293101324543853, 1761723806628841069, 320633732806449074559, 67333083889354305657391, 16159940133445033357773841, 4395503716297049073314484753

Offset: 0

Seiichi Manyama, Jan 05 2024

nonn

Cf. A010790, A066998, A228229.

a(n) = (n+1)*n!^2*sum(k=1, n, 1/((k+1)*k!^2));

a(0) = 0; a(n) = (n+1) * n * a(n-1) + 1.

a(n) = A228229(n) - (n+1) * (n!)^2.

cp's OEIS Frontend

A215943 Number of ways to place k non-attacking bishops on an n x n toroidal chessboard, summed over all k >= 0.

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A258213 Number of permutations of {1,2,3,...,n} such that no even numbers are adjacent.

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A369597 a(n) is the number of reduced stable marriage problem instances of order 3 that generate n possible stable matchings.

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A084259 a(n) = n! * (n+1)! * (n+2)! * (n+3).

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A176049 Primes of the form n!*(n+1)!*(n+2)! - 1 or n!*(n+1)!*(n+2)! + 1.

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A258216 Number of permutations of {1,2,3,...,n} such that no multiples of 3 are adjacent.

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A273935 Number of ways to arrange n women and n men around a circular table so that they can be divided into n nonintersecting pairs of 1 woman and 1 man sitting side-by-side.

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A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

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A362588 Triangle read by rows, T(n, k) = RisingFactorial(n - k, k) * FallingFactorial(n, k).

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A176049 Primes of the form n!(n+1)!(n+2)! - 1 or n!(n+1)!(n+2)! + 1.