A238258 -id:A238258 - cp's OEIS frontend

A002465 Number of ways to place n nonattacking bishops on an n X n board.

1, 1, 4, 26, 260, 3368, 53744, 1022320, 22522960, 565532992, 15915225216, 496911749920, 17029582652416, 636101065346560, 25705530908501760, 1118038500044633088, 52054862490790200576, 2584158975023147147264

Offset: 0

N. J. A. Sloane

The old name of this sequence was wrong. It was corrected by Vaclav Kotesovec, Feb 19 2011. Kotesovec remarks that the maximal number of nonattacking bishops on an n X n board is 2n-2, and there are 2^n ways to place them. See the Kotesovec link.

			a(3) = 26: ways to place 3 nonattacking bishops on a 3 X 3 board:
  XXX XXO XXO XOX OXO
  OOO OOO OOO OOO OXO
  OOO XOO OXO OXO OXO
  (4) (8) (8) (4) (2)

W. Ahrens, Mathematische Unterhaltungen und Spiele. Teubner, Leipzig, Vol. 1, 3rd ed., 1921; Vol. 2, 2nd ed., 1918. See Vol. 1, p. 271.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Vilenkin, Populyarnaja kombinatorika, 1972, p. 166.

Vaclav Kotesovec, Table of n, a(n) for n = 0..375
W. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig: B. G. Teubner, 1901.
S. E. Arshon, Solution of one combinatorial problem [in Russian], Matematicheskoe prosveshchenie, Ser. 1, 8, 1936, pp. 24-29.
D. Atkinson, Solution to the n-Bishops problem of trying to place n identical bishops on an n x n chessboard. [Broken link?]
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 242-252.
J. Perott, Sur le problème des fous, Bulletin de la société mathématique de France, Tome XI, 1883, p. 173-186.
Eric Weisstein's World of Mathematics, Bishops Problem.

Cf. A238258, A238260, A187235.

Main diagonal of A378590.

peven[i_]:=(Sum[(-1)^j*Binomial[n-i-1,j]/(n-i-1)!*(n-i+1-j)^(n/2)*(n-i-j)^(n/2-1),{j,0,n-i-1}]);
poddblack[i_]:=(Sum[(-1)^j*Binomial[n-i-1,j]/(n-i-1)!*(n-i+1-j)^((n+1)/2)*(n-i-j)^((n-3)/2),{j,0,n-i-1}]);
poddwhite[i_]:=(Sum[(-1)^j*Binomial[n-i-1,j]/(n-i-1)!*(n-i+1-j)^((n-1)/2)*(n-i-j)^((n-1)/2),{j,0,n-i-1}]);
Table[If[n==1,1,Sum[If[EvenQ[n],peven[i]*peven[n-i],poddblack[i]*poddwhite[n-i]],{i,1,n-1}]],{n,1,50}]
(* Alternative formula with Stirling numbers of the second kind: *)
Table[If[n==1,1, Sum[Sum[Binomial[Floor[(n+1)/2],j] * StirlingS2[j+Floor[n/2],n-i], {j,0,Floor[(n+1)/2]}] * Sum[Binomial[Floor[n/2],j] * StirlingS2[j+Floor[(n+1)/2],i], {j,0,Floor[n/2]}], {i,1,n-1}]], {n,1,50}] (* Vaclav Kotesovec, Mar 23 2011 *)

Asymptotic: a(n)/(n-1)! ~ 0.631266 * 3.08827^n. - Vaclav Kotesovec, Mar 23 2011
The second constant is 2/(z*(2-z)) = 3.0882773047417401791158400820254..., where z is the root z=1.593624260040... of the equation exp(z)*(2-z)=2. - Vaclav Kotesovec, May 27 2011
For constants see A238258 and A238260. - Vaclav Kotesovec, Feb 21 2014

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006
Definition corrected by Vaclav Kotesovec, Feb 19 2011
Terms a(11)-a(17) from Vaclav Kotesovec, Mar 09 2011
a(0)=1 prepended by Alois P. Heinz, Dec 01 2024

A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

Original entry on oeis.org

1, 0, 1, 6, 61, 770, 12160, 228382, 4989621, 124262532, 3475892685, 107901412520, 3681266754660, 136918473752216, 5513911474915116, 239034083286873630, 11098790133822288645, 549539910028075555016, 28903562131933534643851, 1609321474965547356327246

Offset: 0

Vaclav Kotesovec, Feb 28 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008277, A014322, A047797, A342111.

Cf. A048993.

[(&+[StirlingSecond(n, k)*StirlingSecond(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Table[Sum[StirlingS2[n, k]*StirlingS2[n, n-k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, stirling(n, k, 2)*stirling(n, n-k, 2)); \\ Michel Marcus, Feb 28 2021

[sum( stirling_number2(n, k)*stirling_number2(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

From Vaclav Kotesovec, Feb 28 2021, updated May 25 2025: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773047417401791158400820254382768364448971420138767247...
c = 1/(2*Pi*sqrt((1 + LambertW(-2*exp(-2)))*(3 + LambertW(-2*exp(-2))))) = 0.12826577250734152801558828593238744179869387423941684693208180123477... (End)

A363435 Number of partitions of [2n] having exactly n blocks with all elements of the same parity.

Original entry on oeis.org

1, 0, 5, 42, 569, 9470, 191804, 4534502, 122544881, 3721101192, 125331498349, 4634063018948, 186515332107196, 8114659545679752, 379362605925991692, 18961051425453713478, 1008752282616284996865, 56905048753221935350268, 3392250956149146382053539

Offset: 0

Alois P. Heinz, Jun 01 2023

nonn

			a(2) = 5: 13|24, 14|2|3, 1|2|34, 1|23|4, 12|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Partition of a set

Cf. A124424, A363434, A363451.

g:= proc(n) option remember; `if`(n=0, 1, expand(x*
      add(g(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= (n, k)-> coeff(g(n), x, k):
b:= proc(g, u) option remember;
      add(S(g, k)*S(u, k)*k!, k=0..min(g, u))
    end:
T:= proc(n, k) option remember; local g, u; g:= floor(n/2); u:= ceil(n/2);
      add(add(add(binomial(g, i)*S(i, h)*binomial(u, j)*
      S(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)
    end:
a:= n-> T(2*n, n):
seq(a(n), n=0..18);

b[g_, u_] := b[g, u] = Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, Min[g, u]}];
T[n_, k_] := Module[{g, u}, g = Floor[n/2]; u = Ceiling[n/2]; Sum[Sum[Sum[ Binomial[g, i]*StirlingS2[i, h]*Binomial[u, j]*StirlingS2[j, k - h]*b[g - i, u - j], {j, k - h, u}], {i, h, g}], {h, 0, k}]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz in A124424 *)

a(n) = A124424(2n,n).

Conjecture: Limit_{n->oo} (a(n)/n!)^(1/n) = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773... - Vaclav Kotesovec, Oct 21 2023

A238260 Decimal expansion of a multiplicative constant related to A002465.

Original entry on oeis.org

6, 3, 1, 2, 6, 6, 8, 7, 8, 8, 7, 4, 1, 1, 5, 4, 6, 7, 9, 7, 0, 0, 4, 8, 2, 3, 2, 5, 7, 9, 7, 0, 6, 8, 7, 9, 5, 5, 6, 1, 5, 4, 6, 9, 0, 5, 1, 4, 4, 6, 1, 1, 4, 0, 8, 9, 2, 0, 0, 6, 9, 7, 3, 4, 0, 5, 0, 8, 5, 4, 1, 5, 0, 3, 7, 6, 6, 1, 7, 0, 8, 5, 6, 0, 4, 0, 0, 8, 5, 0, 1, 7, 6, 1, 1, 0, 9, 3, 3, 5, 4, 6, 3, 5, 5

Offset: 0

Vaclav Kotesovec, Feb 21 2014

			0.63126687887411546797...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.249

Cf. A002465, A238258, A187235.

Equals lim n->infinity A002465(n) / ((n-1)! * A238258^n).

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A002465 Number of ways to place n nonattacking bishops on an n X n board.

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A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

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A363435 Number of partitions of [2n] having exactly n blocks with all elements of the same parity.

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A238260 Decimal expansion of a multiplicative constant related to A002465.

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