A122747 -id:A122747 - cp's OEIS frontend

A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

1, 1, 1, 1, 3, 8, 36, 110, 666, 3250, 23436, 125198, 1037520, 7241272, 66360960, 500827928, 5080370400, 45926666984, 508032504000, 4919789029480, 59256857923200, 656763542278304, 8532986822438400, 100525959568386848, 1405335514253932800, 18431883489984091552

Offset: 0

N. J. A. Sloane

From Vaclav Kotesovec, Apr 26 2012: (Start)
This sequence gives (according to the article by Robinson) the number of inequivalent solutions.
For the total number of all arrangements of n non-attacking bishops such that every square of the board is controlled by at least one bishop, see A122749.
For the total number of all arrangements of n bishops (in any position) such that every square of the board is controlled by at least one bishop, see A182333.
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..250
Jean-François Alcover, Mathematica program.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Cf. A005631, A005632, A005633, A005634, A122749, A123071, A123072, A182333.

E:=proc(n) local k; if n mod 2 = 0 then k := n/2; if k mod 2 = 0 then RETURN( (k!*(k+2)/2)^2 ); else RETURN( ((k-1)!*(k+1)^2/2)^2 ); fi; else k := (n-1)/2; if k mod 2 = 0 then RETURN( ((k!)^2/12)*(3*k^3+16*k^2+18*k+8) ); else RETURN( ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3) ); fi; fi; end; # Gives A122749
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
C:=proc(n) local k; if n mod 2 = 0 then RETURN(0); fi; k:=(n-1)/2; if k mod 2 = 0 then RETURN( k*2^(k-1)*((k/2)!)^2 ); else RETURN( 2^k*(((k+1)/2)!)^2 ); fi; end; # Gives A122693
Q:=proc(n) local m; if n mod 8 <> 1 then RETURN(0); fi; m:=(n-1)/8; ((2*m)!)^2/(m!)^2; end; # Gives A122747
M:=proc(n) local k; if n mod 2 = 0 then k:=n/2; if k mod 2 = 0 then RETURN( k!*(k+2)/2 ); else RETURN( (k-1)!*(k+1)^2/2 ); fi; else k:=(n-1)/2; RETURN(D(k)*D(k+1)); fi; end; # Gives A122748
a:=n-> if n <= 1 then RETURN(1) else E(n)/8 + C(n)/8 + Q(n)/4 + M(n)/4; fi; # Gives A005635
# The following additional Maple programs produce A123071, A005631, A123072, A005633, A005632, A005634
S:=proc(n) local k; if n mod 2 = 0 then RETURN(0) else k:=(n-1)/2; RETURN(B(k)*B(k+1)); fi; end; # Gives A123071
psi:=n->S(n)/2; # Gives A005631
zeta:=n->Q(n)/2; # Gives A123072
mu:=n->(M(n)-S(n))/2; # Gives A005633
chi:=n->(C(n)-S(n)-Q(n))/4; # Gives A005632
eps:=n->E(n)/8-C(n)/8+S(n)/4-M(n)/4; # Gives A005634

Entry revised by N. J. A. Sloane, Sep 25 2006

A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320

Offset: 0

Philippe Deléham, Oct 28 2011

Unsigned version of A021012.

Equal to A136572*A007318.

			Triangle begins:
    1;
    1,   1;
    2,   4,    2;
    6,  18,   18,    6;
   24,  96,  144,   96,  24;
  120, 600, 1200, 1200, 600, 120;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..495
P. Bala, Deformations of the Hadamard product of power series
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Cf. A007318, A008619, A021012, A084938, A136572.

Columns include: A000142, A001563, A001804, A001805, A001806, A001807.

/* As triangle */ [[Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 28 2015

Table[n!*Binomial[n, j], {n, 0, 30}, {j, 0, n}] (* G. C. Greubel, Sep 27 2015 *)

factorial(n)*binomial(n,k) # Danny Rorabaugh, Sep 27 2015

T(n,k) is given by (1,1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,1,2,2,3,3,4,4,5,5,6,6, ...) where DELTA is the operator defined in A084938.
Sum_{k>=0} T(m,k)*T(n,k) = (m+n)!.
T(2n,n) = A122747(n).
Sum_{k>=0} T(n,k)^2 = A010050(n) = (2n)!.
Sum_{k>=0} T(n,k)*x^k = A000007(n), A000142(n), A000165(n), A032031(n), A047053(n), A052562(n), A047058(n), A051188(n), A051189(n), A051232(n), A051262(n), A196258(n), A145448(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
The row polynomials have the form (x + 1) o (x + 2) o ... o (x + n), where o denotes the black diamond multiplication operator of Dukes and White. See example E10 in the Bala link. - Peter Bala, Jan 18 2018

Name exchanged with a formula by Peter Luschny, Feb 01 2015

A320958 The exponential limit of arcsin (odd indices only).

Original entry on oeis.org

1, 5, 468, 197325, 233145675, 605979974250, 2987147975582925, 25254853526009732625, 340477692051264295027500, 6926101229658271208893970625, 203562520854789108487169894574375, 8346651541805126492397454664310896250, 463877742240727904202821053051014479795625

Offset: 0

Peter Luschny, Nov 08 2018

nonn

See A320956 for definitions and comments.

			Illustration of the convergence in the sense of A320956:
   [0] 0, 0, 0, 0, 0,   0, 0,      0, 0,         0, ...
   [1] 0, 1, 0, 1, 0,   9, 0,    225, 0,     11025, ... A177145, A001818
   [2] 0, 1, 0, 4, 0, 144, 0,  14400, 0,   2822400, ... A122747
   [3] 0, 1, 0, 5, 0, 369, 0,  82125, 0,  36173025, ...
   [4] 0, 1, 0, 5, 0, 459, 0, 160875, 0, 121837275, ...
   [5] 0, 1, 0, 5, 0, 468, 0, 192375, 0, 198472050, ...
   [6] 0, 1, 0, 5, 0, 468, 0, 197100, 0, 227644200, ...
   [7] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 232737750, ...
   [8] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 233134650, ...
   [9] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 233145675, ...

Cf. A320955 (exp), A320962 (log(x+1)), A320956 (sec+tan), this sequence (arcsin),
A320959 (arctanh).
Cf. A177145, A001818, A122747.

# Function ExpLim defined in A320956.
L := [ExpLim(28, arcsin)]: seq(L[2*n], n=1..13);

m = 13; CoefficientList[ArcSin[x] + O[x]^(2 m + 1), x]*Range[0, 2 m - 1]!*BellB[Range[0, 2 m - 1]] // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 23 2019 *)

A072477 a(n) = (2n)!binomial(2*n,n)/8.

Original entry on oeis.org

18, 1800, 352800, 114307200, 55324684800, 37399486924800, 33659538232320000, 38910426196561920000, 56186655427835412480000, 99113260174701667614720000, 209723658529668728672747520000, 524309146324171821681868800000000, 1528885470681285032024329420800000000

Offset: 2

N. J. A. Sloane, Aug 02 2002

Michel Bousquet, Gilbert Labelle and Pierre Leroux, Enumeration of planar two-face maps, Discrete Mathematics, 222 (2000), 1-25.

Cf. A000984, A122747.

a[n_] := (2*n)!*Binomial[2*n, n]/8; Array[a, 12, 2] (* Amiram Eldar, Jun 18 2025 *)

a(n) = (2*n)!*binomial(2*n, n)/8; \\ Amiram Eldar, Jun 18 2025

From Amiram Eldar, Jun 18 2025: (Start)
Sum_{n>=2} 1/a(n) = 2*Pi*L_0(1/2) - 2, where L is the modified Struve function.
Sum_{n>=2} (-1)^n/a(n) = 2 - 2*Pi*H_0(1/2), where H is the Struve function. (End)

A360602 a(n) = ((2*n + 1)! / n!)^2 / (n + 1).

Original entry on oeis.org

1, 18, 1200, 176400, 45722880, 18441561600, 10685567692800, 8414884558080000, 8646761377013760000, 11237331085567082496000, 18020592759036666839040000, 34953943088278121445457920000, 80662945588334126412595200000000, 218412210097326433146332774400000000

Offset: 0

Peter Luschny, Feb 16 2023

nonn

Eric Weisstein's World of Mathematics, Modified Struve Function.
Eric Weisstein's World of Mathematics, Struve function.
Wikipedia, Struve function.

Cf. A122747, A000108, A000407.

a := n -> ((2*n + 1)! / n!)^2 / (n + 1):
seq(a(n), n = 0..13);

a[n_] := ((2*n + 1)!/n!)^2/(n + 1); Array[a, 14, 0] (* Amiram Eldar, Mar 02 2023 *)

a(n) = CatalanNumber(n) * (2*n)! * (2*n + 1)^2.
a(n) ~ (4*n/e)^(2*n)*(8*n - 2/3).
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/2 + StruveL(0, 1/2)*Pi/2 + StruveL(1, 1/2)*Pi/4, where StruveL is the modified Struve function.
Sum_{n>=0} (-1)^n/a(n) = 1/2 + StruveH(0, 1/2)*Pi/2 - StruveH(1, 1/2)*Pi/4, where StruveH is the Struve function. (End)

cp's OEIS Frontend

A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

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A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

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A320958 The exponential limit of arcsin (odd indices only).

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A072477 a(n) = (2*n)!*binomial(2*n,n)/8.

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A360602 a(n) = ((2*n + 1)! / n!)^2 / (n + 1).

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A072477 a(n) = (2n)!binomial(2*n,n)/8.