A238261 -id:A238261 - cp's OEIS frontend

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

1, 5, 51, 769, 15345, 381065, 11323991, 391861841, 15476988033, 687029386845, 33861652925595, 1834814222811361, 108411291759763681, 6936921762461326545, 477881176664541171375, 35264213540563039871265, 2775185864375851234241985, 232010235620834821000259765, 20534530616200868936398461635

Offset: 1

Vaclav Kotesovec, Mar 08 2011

Two semi-bishops do not attack each other if they are in the same NorthWest-SouthEast diagonal.

Conjecture: Number of parity preserving permutations of the set {1, 2, ..., 2n+1} with exactly n+1 cycles (see A246117). - Peter Luschny, Feb 09 2015

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 260-265.

Cf. A238261, A238262, A002465, A099152, A000255, A129256.

Table[If[n==1,1,Coefficient[Expand[Product[x+i,{i,1,n}]*Product[x+i,{i,1,n-1}],x],x,n-1]],{n,1,50}]
Table[(-1)^n*Sum[StirlingS1[n+1,j]*StirlingS1[n,n-j+1],{j,1,n}],{n,1,50}] (* Explicit formula, Vaclav Kotesovec, Mar 24 2011 *)

a(n) = {(-1)^n*sum(i=0, n, stirling(n,i,1) * stirling(n+1,n-i+1,1))} \\ Andrew Howroyd, May 09 2020

a(n)/(n-1)! ~ 0.24252191 * 4.9108149^n where the second constant is 1/(z*(1-z)) = 4.910814964..., where z=0.715331862959... is a root of the equation z=2*(z-1)*log(1-z).
For constants see A238261 and A238262. - Vaclav Kotesovec, Feb 21 2014
a(n) = (-1)^n * Sum_{i=0..n} Stirling1(n,i) * Stirling1(n+1,n-i+1). - Ryan Brooks, May 09 2020

A129256 Central coefficient of Product_{k=0..n} (1+k*x)^2.

Original entry on oeis.org

1, 2, 13, 144, 2273, 46710, 1184153, 35733376, 1251320145, 49893169050, 2232012515445, 110722046632560, 6032418472347265, 358103844593876654, 23007314730623658225, 1590611390957425536000, 117745011140615270168865

Offset: 0

Paul D. Hanna, Apr 06 2007

nonn

			This sequence equals the central terms of the triangle in which the g.f. of row n is (1+x)^2*(1+2x)^2*(1+3x)^2*...*(1+n*x)^2, as illustrated by:
  (1);
   1, (2),  1;
   1,  6, (13),  12,     4;
   1, 12,  58, (144),  193,    132,      36;
   1, 20, 170,  800, (2273),  3980,    4180,   2400,    576;
   1, 30, 395, 3000, 14523, (46710), 100805, 143700, 129076, 65760, 14400;
  ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..354

Cf. A008275 (Stirling1 numbers), A187235, A238261, A246117, A254882, A350376.

Cf. A384012, A384031, A384017.

Flatten[{1,Table[Coefficient[Expand[Product[(1+k*x),{k,0,n}]^2],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

PARI
```
a(n)=polcoeff(prod(k=0,n,1+k*x)^2,n)
```

{a(n)=(-1)^n*sum(k=0,n,stirling(n+1,k+1,1)*stirling(n+1,n-k+1,1))} \\ Paul D. Hanna, Jul 16 2009

a(n) = (-1)^n*Sum_{k=0..n} Stirling1(n+1,k+1)*Stirling1(n+1,n-k+1). - Paul D. Hanna, Jul 16 2009

a(n) ~ c * d^n * (n-1)!, where d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.910814964568255..., c = (-LambertW(-1, -exp(-1/2)/2))^(3/2)/(sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.851946112888790982829578047527831525434714038256... . - Vaclav Kotesovec, Feb 10 2015, updated May 14 2025

A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

Original entry on oeis.org

1, 0, 1, 12, 193, 3980, 100805, 3034920, 105994833, 4215106728, 188097696345, 9309515255700, 506149663220641, 29989851619249236, 1923467938147053389, 132771455705186298000, 9814431285244231295265, 773520674985391641371280, 64752473306596841023424945

Offset: 0

Vaclav Kotesovec, Feb 28 2021

nonn

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

Cf. A008275, A014322, A047796, A132393, A342110.
Cf. A048994, A155826.
Cf. A129256, A187235, A246117.

[(&+[(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Table[(-1)^n*Sum[StirlingS1[n, k]*StirlingS1[n, n-k], {k, 0, n}], {n, 0, 20}]

a(n) = (-1)^n*sum(k=0, n, stirling(n, k, 1)*stirling(n, n-k, 1)); \\ Michel Marcus, Feb 28 2021

[sum( stirling_number1(n, k)*stirling_number1(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

a(n) ~ c * d^n * (n-1)!, where
d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.9108149645682558987515348052403521978987052817678471761394112...
c = 1/(4*sqrt(-LambertW(-1, -exp(-1/2)/2)) * sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.06903826111269387517867145566264007373042059749428879149076344304196548... - Vaclav Kotesovec, Feb 28 2021, updated May 14 2025
a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^2. - Seiichi Manyama, May 13 2025

A330287 Permanent of the n-th principal submatrix M(n) of A319840.

Original entry on oeis.org

1, 1, 8, 208, 11488, 1093056, 158972160, 32734095360, 9049229328384, 3230305304002560, 1445344680438005760, 791762592707031859200, 521023492500173338705920, 405448567547957922512240640, 368210800911998093644372377600, 385879616532879866123928993792000, 462151848929747968377341029122048000

Offset: 0

Stefano Spezia, Dec 11 2019

nonn

The matrix M(n) is defined as M[i,j,n] = i*j if i < 3 or j < 3 and M[i,j,n] = 2*(i + j) - 4 otherwise.
det(M(0)) = det(M(1)) = 1 and det(M(n)) = 0 for n > 1.
For n > 0, the trace of the matrix M(n) is A001844(n-1).
For n > 0, the antitrace of the matrix M(n) is A005893(n-1).
For n > 1, the super- and subdiagonal sum is A001105(n-1).

			For n = 1 the matrix M(1) is
  1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
  1, 2
  2, 4
with permanent a(2) = 8.
For n = 3 the matrix M(3) is
  1,  2,  3
  2,  4,  6
  3,  6,  8
with permanent a(3) = 208.

Vaclav Kotesovec, Table of n, a(n) for n = 0..35

Cf. A001105, A001844, A005893, A319840.

tm(n) = matrix(n, n, i, j, if ((i<3) || (j<3), i*j, 2*(i+j)-4));
a(n) = matpermanent(tm(n));

a(n) ~ c * A238261^n * n!^2 / sqrt(n), where c = 0.0286685259829... - Vaclav Kotesovec, Aug 19 2021

A238262 Decimal expansion of a multiplicative constant related to A187235.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Feb 21 2014

			0.242521912815235985949321...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.261.

Cf. A187235, A238261, A002465.

Equals lim n->infinity A187235(n) / ((n-1)! * A238261^n).

cp's OEIS Frontend

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

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A129256 Central coefficient of Product_{k=0..n} (1+k*x)^2.

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A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

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A330287 Permanent of the n-th principal submatrix M(n) of A319840.

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A238262 Decimal expansion of a multiplicative constant related to A187235.

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