A002465 -id:A002465 - cp's OEIS frontend

A007820 Stirling numbers of second kind S(2n,n).

1, 1, 7, 90, 1701, 42525, 1323652, 49329280, 2141764053, 106175395755, 5917584964655, 366282500870286, 24930204590758260, 1850568574253550060, 148782988064375309400, 12879868072770626040000, 1194461517469807833782085, 118144018577011378596484455

Offset: 0

kemp(AT)sads.informatik.uni-frankfurt.de (Rainer Kemp)

Chan and Manna prove that a(n) is odd if and only if n is in A003714. - Jason Kimberley, Sep 14 2009
The number of ways to partition a set of 2*n elements into n disjoint subsets. - Vladimir Reshetnikov, Oct 10 2016
Conjecture: a(2*n+1) is divisible by (2*n + 1)^2. - Peter Bala, Mar 30 2025

			G.f.: A(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 +...,
where A(x) = 1 + 1^2*x*exp(-1*x) + 2^4*exp(-2^2*x)*x^2/2! + 3^6*exp(-3^2*x)*x^3/3! + 4^8*exp(-4^2*x)*x^4/4! + 5^10*exp(-5^2*x)*x^5/5! + ... - _Paul D. Hanna_, Oct 17 2012

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

Alois P. Heinz, Table of n, a(n) for n = 0..345 (terms n = 1..200 from Vincenzo Librandi)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O-Y. Chan and D. V. Manna, Divisibility properties of Stirling numbers of the second kind [From _Jason Kimberley_, Sep 14 2009]

Cf. A002465, A008277, A187646, A191236, A217913, A217914, A217915, A247238.

Cf. A350376, A383862, A384060.

A007820 := proc(n) Stirling2(2*n,n) ; end proc:
seq(A007820(n),n=0..20) ; # R. J. Mathar, Mar 15 2011

Table[StirlingS2[2n, n], {n, 1, 12}] (* Emanuele Munarini, Mar 12 2011 *)

makelist(stirling2(2*n,n),n,0,12); /* Emanuele Munarini, Mar 12 2011 */

a(n)=stirling(2*n,n,2); /* Joerg Arndt, Jul 01 2011 */

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), n)} \\ Paul D. Hanna, Oct 17 2012

{a(n)=polcoeff(sum(m=1,n,(m^2)^m*exp(-m^2*x+x*O(x^n))*x^m/m!),n)} \\ Paul D. Hanna, Oct 17 2012

from sympy.functions.combinatorial.numbers import stirling
def A007820(n): return stirling(n<<1,n) # Chai Wah Wu, Jun 09 2025

[stirling_number2(2*i,i) for i in range(1,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = A048993(2n,n). - R. J. Mathar, Mar 15 2011
Asymptotic: a(n) ~ (4*n/(e*z*(2-z)))^n/sqrt(2*Pi*n*(z-1)), where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2. - Vaclav Kotesovec, May 30 2011
a(n) = 1/n! * Sum_{k = 0..n} binomial(n,k)*(-1)^k*(n-k)^(2*n). - Emanuele Munarini, Jul 01 2011
a(n) = [x^n] 1 / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Oct 17 2012
O.g.f.: Sum_{n>=1} (n^2)^n * exp(-n^2*x) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n. - Paul D. Hanna, Oct 17 2012
G.f.: Sum_{n > 0} (a(n)*n!/(2*n)!)*x^n = x*B'(x)/B(x)-1, where B(x) satisfies B(x)^2 = x*(exp(B(x))-1). - Vladimir Kruchinin, Mar 13 2013
a(n) = Sum_{j = 0..n} (-1)^(n-j)*n^j*binomial(2*n,j)*stirling2(2*n-j,n). - Vladimir Kruchinin, Jun 14 2013

Typo in Mathematica program fixed by Vincenzo Librandi, May 04 2013

a(0)=1 prepended by Alois P. Heinz, Feb 01 2018

A137774 Number of ways to place n nonattacking empresses on an n X n board.

Original entry on oeis.org

1, 2, 2, 8, 20, 94, 438, 2766, 19480, 163058, 1546726, 16598282, 197708058, 2586423174, 36769177348, 563504645310, 9248221393974, 161670971937362, 2996936692836754, 58689061747521430, 1210222434323163704, 26204614054454840842, 594313769819021397534, 14086979362268860896282

Offset: 1

Vaclav Kotesovec, Jan 27 2011

An empress moves like a rook and a knight.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Separators - a new statistic for permutations, arXiv:1905.12364 [math.CO], 2019.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, On the Sparseness of the Downsets of Permutations via Their Number of Separators, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 3, Article #S2R21.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.685 and 636.
W. Schubert, N-Queens page

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774, A245011.
Cf. A218244, A002464, A110128, A117574, A089222, A002493.

Asymptotics (Vaclav Kotesovec, Jan 26 2011): a(n)/n! -> 1/e^4.
General asymptotic formulas for number of ways to place n nonattacking pieces rook + leaper[r,s] on an n X n board:
a(n)/n! -> 1/e^2 for 0a(n)/n! -> 1/e^4 for 0

Terms a(16)-a(17) from Vaclav Kotesovec, Feb 06 2011
Terms a(18)-a(19) from Wolfram Schubert, Jul 24 2011
Terms a(20)-a(24) (computed by Wolfram Schubert), Vaclav Kotesovec, Aug 25 2012

A274105 Triangle read by rows: T(n,k) = number of configurations of k nonattacking bishops on the black squares of an n X n chessboard (0 <= k <= n - [n>1]).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 5, 4, 1, 8, 14, 4, 1, 13, 46, 46, 8, 1, 18, 98, 184, 100, 8, 1, 25, 206, 674, 836, 308, 16, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 41, 612, 4196, 13756, 20476, 11896, 1912, 32, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 61, 1440, 16940, 106772, 361780, 629336, 506600, 154256, 11600, 64

Offset: 0

N. J. A. Sloane, Jun 14 2016

Rows give the coefficients of the independence polynomial of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017

			Triangle begins:
  1;
  1,  1;
  1,  2;
  1,  5,   4;
  1,  8,  14,      4;
  1, 13,  46,     46,      8;
  1, 18,  98,    184,    100,      8;
  1, 25,  206,   674,    836,    308,     16;
  1, 32,  356,  1704,   3532,   2816,    632,     16;
  1, 41,  612,  4196,  13756,  20476,  11896,   1912,     32;
  1, 50,  940,  8480,  38932,  89256,  93800,  37600,   3856,    32;
  1, 61, 1440, 16940, 106772, 361780, 629336, 506600, 154256, 11600, 64;
  ...
Corresponding independence polynomials:
  1, (empty graph)
  1+x, (K_1)
  1+2*x, (P_2 = K_2)
  1+5*x+4*x^2, (butterfly graph)
  1+8*x+14*x^2+4*x^3,
  ...

Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. See Section 9.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. See Section 9. [Annotated scanned copy]
Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas. arXiv:2411.16492 [math.CO], 2024. (considered as white board).
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Independence Polynomial

Alternate rows give A088960.
Row sums are A216332(n+1).
Cf. A274106 (white squares), A288183, A201862, A002465.

with(combinat); with(gfun);
T:=n->add(stirling2(n+1,n+1-k)*x^k, k=0..n);
# bishops on black squares
bish:=proc(n) local m,k,i,j,t1,t2; global T;
if n<2 then return [1$(n+1)] fi;
if (n mod 2) = 0 then m:=n/2;
t1:=add(binomial(m,k)*T(2*m-1-k)*x^k, k=0..m);
else
m:=(n-1)/2;
t1:=add(binomial(m+1,k)*T(2*m-k)*x^k, k=0..m+1);
fi;
seriestolist(series(t1,x,2*n+1));
end;
for n from 0 to 12 do lprint(bish(n)); od:
# second Maple program:
T:= (n,k)-> add(binomial(ceil(n/2),j)*Stirling2(n-j,n-k),j=0..k):
seq(seq(T(n,k), k=0..n-`if`(n>1,1,0)), n=0..11);  # Alois P. Heinz, Dec 01 2024

CoefficientList[Table[Sum[x^n Binomial[Ceiling[n/2], k] BellB[n - k, 1/x], {k, 0, Ceiling[n/2]}], {n, 10}], x] (* Eric W. Weisstein, Jun 26 2017 *)

def stirling2_negativek(n, k):
  if k < 0: return 0
  else: return stirling_number2(n, k)
print([sum([binomial(ceil(n/2), l)*stirling2_negativek(n-l, n-k) for l in [0..k]]) for n in [0..10] for k in [0..n-1+kronecker_delta(n,1)+kronecker_delta(n,0)]]) # Eder G. Santos, Dec 01 2024

From Eder G. Santos, Dec 01 2024: (Start)
T(n,k) = Sum_{j=0..k} binomial(ceiling(n/2),j) * Stirling2(n-j,n-k).
T(n,k) = T(n-1,k) + (n-k+A000035(n)) * T(n-1,k-1), T(n,0) = 1, T(0,k) = delta(k,0). (End)

T(0,0) prepended by Eder G. Santos, Dec 01 2024

A201861 Number of ways to place n nonattacking ferses on an n X n board.

Original entry on oeis.org

1, 4, 38, 661, 16286, 527654, 21191208, 1015335608, 56484795166, 3576188894116, 253756155257774, 19937566770720487, 1717714713900798962, 160977153444563000938, 16300053518916522372836, 1773133639291617644092637, 206197950879511078156507433

Offset: 1

Vaclav Kotesovec, Dec 06 2011

Fers is a leaper [1,1].

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 423.

Cf. A201243 - A201248, A002465, A067965.

Asymptotic (Kotesovec, 2011): a(n) ~ n^(2n)/n!*exp(-5/2).

a(15) from Vaclav Kotesovec, Jan 03 2012
a(16) from Vaclav Kotesovec, Aug 31 2016
a(17) from Vaclav Kotesovec, May 30 2021

A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

Original entry on oeis.org

1, 2, 9, 70, 729, 9918, 167281, 3423362, 82609921, 2319730026, 74500064809, 2711723081550, 110568316431609, 5016846683306758, 251180326892449969, 13806795579059621930, 827911558468860287041, 53940895144894708523922, 3799498445458163685753481, 288400498147873552894868886

Offset: 0

Vaclav Kotesovec, Dec 06 2011

nonn

Also the number of vertex covers and independent vertex sets of the n X n bishop graph.

Vaclav Kotesovec, Table of n, a(n) for n = 0..320
Vaclav Kotesovec, Non-attacking chess pieces
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A010790, A172123, A172124, A172127, A172129, A176886, A187239 - A187242, A002465, A216078, A216332.

Row sums of A378590.

knbishops[k_,n_]:=(If[n==1,If[k==1,1,0],(-1)^k/(2n-k)!
*Sum[Binomial[2n-k,n-k+i]*Sum[(-1)^m*Binomial[n-i,m]*m^Floor[n/2]*(m+1)^Floor[(n+1)/2],{m,1,n-i}]
*Sum[(-1)^m*Binomial[n-k+i,m]*m^Floor[(n+1)/2]*(m+1)^Floor[n/2],{m,1,n+i-k}],{i,Max[0,k-n],Min[k,n]}]]);
Table[1+Sum[knbishops[k,n],{k,1,2n-1}],{n,1,25}]

a(n) = A216078(n+1) * A216332(n+1). - Andrew Howroyd, May 08 2017

a(0)=1 prepended by Alois P. Heinz, Dec 01 2024

A187239 Number of ways to place 7 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 440, 38368, 1022320, 14082528, 126490352, 837543200, 4412818240, 19447224864, 74255991784, 251997948736, 774861621936, 2191005028672, 5764306674400, 14243327787456, 33309659739904, 74194554880960, 158241369977880, 324605935279648, 642894402918768

Offset: 1

Vaclav Kotesovec, Mar 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
E. Weisstein, Bishops Problem, mathWorld.
Index entries for linear recurrences with constant coefficients, signature (6, -6, -34, 84, 42, -322, 162, 603, -708, -540, 1260, 0, -1260, 540, 708, -603, -162, 322, -42, -84, 34, 6, -6, 1).

Cf. A172123, A172124, A172127, A172129, A176886.

CoefficientList[Series[- 8 x^4 (630 x^18 + 10620 x^17 + 153525 x^16 + 1211058 x^15 + 6621390 x^14 + 24647178 x^13 + 66958554 x^12 + 133891418 x^11 + 202680754 x^10 + 232634698 x^9 + 204008900 x^8 + 135332502 x^7 + 67245306 x^6 + 24326718 x^5 + 6174582 x^4 + 1024222 x^3 + 99344 x^2 + 4466 x + 55) / ((x - 1)^15 (x + 1)^9), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = n^14/5040 - n^13/180 + 313n^12/4320 - 383n^11/648 + 14797n^10/4320 - 38233n^9/2520 + 3217n^8/60 - 145469n^7/945 + 1546679n^6/4320 - 4297801n^5/6480 + 257903n^4/270 - 3915679n^3/3780 + 1787007n^2/2240 - 318023n/840 + 9503/128 + (-n^8/192 + n^7/8 - 389n^6/288 + 689n^5/80 - 319n^4/9 + 1153n^3/12 - 95965n^2/576 + 20129n/120 - 9503/128)*(-1)^n.
G.f.: -8x^5*(630x^18 + 10620x^17 + 153525x^16 + 1211058x^15 + 6621390x^14 + 24647178x^13 + 66958554x^12 + 133891418x^11 + 202680754x^10 + 232634698x^9 + 204008900x^8 + 135332502x^7 + 67245306x^6 + 24326718x^5 + 6174582x^4 + 1024222x^3 + 99344x^2 + 4466x + 55)/((x-1)^15*(x+1)^9).
a(7) = A002465(7).

A191236 Number of ways to place n nonattacking bishops on black squares of a 2n X 2n board.

Original entry on oeis.org

1, 2, 14, 184, 3532, 89256, 2800016, 104967808, 4578528464, 227816059360, 12735645181536, 790296855912576, 53905019035510528, 4008716449677965312, 322807879692969879552, 27983800239966141382656

Offset: 0

Vaclav Kotesovec, May 27 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..340
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 253.

Cf. A002465, A187235, A217905.

Join[{1}, Table[(1/n!)*Sum[(-1)^(n - k)*Binomial[n, k]*(k*(k + 1))^n, {k, 0, n}], {n,1,50}]] (* G. C. Greubel, Feb 03 2017 *)

{a(n)=polcoeff(sum(m=0,n,m^m*(m+1)^m*x^m*exp(-m*(m+1)*x+x*O(x^n))/m!),n)} \\ Paul D. Hanna, Oct 15 2012

{a(n)=sum(k=0,n, binomial(n,k) * stirling(2*n-k,n,2))} \\ Paul D. Hanna, Nov 13 2012

a(n) = 1/n! * Sum_{j=0..n} (-1)^(n-j) * binomial(n,j) * (j*(j+1))^n.
Asymptotic: a(n) ~ 1/sqrt(Pi*(z-1)*(2-z)*n)*(2*n*exp(z-1)/z)^n or a(n) ~ exp(z/2)*Stirling2(2*n,n) where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2.
O.g.f.: Sum_{n>=0} n^n*(n+1)^n * exp(-n*(n+1)*x) * x^n/n! = Sum_{n>=0} a(n)*x^n. - Paul D. Hanna, Oct 15 2012
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(2*n-k,n), where Stirling2(n,k) = A008277(n,k). - Paul D. Hanna, Nov 13 2012

Offset changed to 0 and a(0)=1 added by Paul D. Hanna, Nov 13 2012

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A238258 Decimal expansion of a constant related to A002465.

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A238260 Decimal expansion of a multiplicative constant related to A002465.

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A007820 Stirling numbers of second kind S(2n,n).

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A187235 Number of ways to place n nonattacking semi-bishops on an n X n board.

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A137774 Number of ways to place n nonattacking empresses on an n X n board.

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A274105 Triangle read by rows: T(n,k) = number of configurations of k nonattacking bishops on the black squares of an n X n chessboard (0 <= k <= n - [n>1]).

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A201861 Number of ways to place n nonattacking ferses on an n X n board.

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