A274105 -id:A274105 - cp's OEIS frontend

A036563 a(n) = 2^n - 3.

-2, -1, 1, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 16381, 32765, 65533, 131069, 262141, 524285, 1048573, 2097149, 4194301, 8388605, 16777213, 33554429, 67108861, 134217725, 268435453, 536870909, 1073741821, 2147483645

Offset: 0

N. J. A. Sloane

a(n+1) is the n-th number with exactly n 1's in binary representation. - Reinhard Zumkeller, Mar 06 2003
Berstein and Onn: "For every m = 3k+1, the Graver complexity of the vertex-edge incidence matrix of the complete bipirtite graph K(3,m) satisfies g(m) >= 2^(k+2)-3." - Jonathan Vos Post, Sep 15 2007
Row sums of triangle A135857. - Gary W. Adamson, Dec 01 2007
a(n) = A164874(n-1,n-2) for n > 2. - Reinhard Zumkeller, Aug 29 2009
Starting (1, 5, 13, ...) = eigensequence of a triangle with A016777: (1, 4, 7, 10, ...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175655. For the central square just one A[5] vector, with decimal value 186, leads to this sequence (n >= 2). For the corner squares this vector leads to the companion sequence A123203. - Johannes W. Meijer, Aug 15 2010
First differences of A095264: A095264(n+1) - A095264(n) = a(n+2). - J. M. Bergot, May 13 2013
a(n+2) is given by the sum of n-th row of triangle of powers of 2: 1; 2 1 2; 4 2 1 2 4; 8 4 2 1 2 4 8; ... - Philippe Deléham, Feb 24 2014
Also, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508. - Robert Price, Mar 09 2017
a(n+3) is the value of the Ackermann function A(3,n) or ack(3,n). - Olivier Gérard, May 11 2018

			a(2) = 1;
a(3) = 2 + 1 + 2 = 5;
a(4) = 4 + 2 + 1 + 2 + 4 = 13;
a(5) = 8 + 4 + 2 + 1 + 2 + 4 + 8 = 29; etc. - _Philippe Deléham_, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Yael Berstein and Shmuel Onn, The Graver complexity of integer programming, Annals of Combinatorics, Vol. 13, No. 3 (2009), pp. 289-296; arXiv preprint, arXiv:0709.1500 [math.CO], 2007.
L' Education Mathématique, Problème 8907, 49e Annee, No 14, 15 Avril 1947, p. 113
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. Sequence is on page 267.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Row sums of triangular array A027960. A column of A119725.
Cf. A081118, A130459, A131112, A050414, A050415, A135857, A000079, A016777, A283508.
Cf. A074877, A304370, A304371, A331372.

List([0..40], n-> 2^n -3); # G. C. Greubel, Nov 18 2019

[2^n-3: n in [0..40]]; // Vincenzo Librandi, May 09 2011

A036563:=n->2^n-3; seq(A036563(n), n=0..40); # Wesley Ivan Hurt, Jun 26 2014

Table[2^n - 3, {n, 0, 40}] (* Wesley Ivan Hurt, Jun 26 2014 *)
LinearRecurrence[{3,-2},{-2,-1},40] (* Harvey P. Dale, Sep 26 2018 *)

a(n)= 2^n-3 \\ Charles R Greathouse IV, Dec 22 2011

def A036563(n): return (1<Chai Wah Wu, Sep 27 2024

[gaussian_binomial(n,1,2)-2 for n in range(0,40)] # Zerinvary Lajos, May 31 2009

a(n) = 2*a(n-1) + 3.
The sequence 1, 5, 13, ... has a(n) = 4*2^n-3. These are the partial sums of A151821. - Paul Barry, Aug 25 2003
a(n) = A118654(n-3, 6), for n > 2. - N. J. A. Sloane, Sep 29 2006
Row sums of triangle A130459 starting (1, 5, 13, 29, 61, ...). - Gary W. Adamson, May 26 2007
Row sums of triangle A131112. - Gary W. Adamson, Jun 15 2007
Binomial transform of [1, 4, 4, 4, ...] = (1, 5, 13, 29, 61, ...). - Gary W. Adamson, Sep 20 2007
a(n) = 2*StirlingS2(n,2) - 1, for n > 0. - Ross La Haye, Jul 05 2008
a(n) = A000079(n) - 3. - Omar E. Pol, Dec 21 2008
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-2*x) - 3/(1-x).
E.g.f.: exp(2*x) - 3*exp(x). (End)
For n >= 3, a(n) = 2<+>n, where operation <+> is defined in A206853. - Vladimir Shevelev, Feb 17 2012
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1, a(0)=-2, a(1)=-1. - Philippe Deléham, Dec 23 2013
Sum_{n>=1} 1/a(n) = A331372. - Amiram Eldar, Nov 18 2020

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 73, 29, 1, 1, 61, 301, 301, 61, 1, 1, 125, 1081, 2069, 1081, 125, 1, 1, 253, 3613, 11581, 11581, 3613, 253, 1, 1, 509, 11593, 57749, 95401, 57749, 11593, 509, 1, 1, 1021, 36301, 268381, 673261, 673261, 268381, 36301, 1021, 1

Offset: 2

N. J. A. Sloane, May 07 2016

Gives number of bitriangular permutations. Could be prefixed with row 0 containing a single 1. - N. J. A. Sloane, Jan 10 2018

			Triangle begins:
n\m  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]  1;
[3]  1,      1;
[4]  1,      5,      1;
[5]  1,     13,     13,      1;
[6]  1,     29,     73,     29,      1;
[7]  1,     61,    301,    301,     61,      1;
[8]  1,    125,   1081,   2069,   1081,    125,      1;
[9]  1,    253,   3613,  11581,  11581,   3613,    253,      1;
...

Gheorghe Coserea, Rows n = 2..101, flattened
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012.
Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n,k}.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
D. E. Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (16.2).
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
J. Riordan, Letter to N. J. A. Sloane, Dec. 1976.

Column 2 is A036563.
Largest term in each row gives A272645.
Second diagonal from the right is 2^i - 3.
Third diagonal from the right edge is A006230.
T(2n,n) gives A048144.
For row sums see A297195.
Cf. A008277, A001469, A371761.

A272644 := proc(n,m)
    add(combinat[stirling2](m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!,i=0..m) ;
end proc:
seq(seq(A272644(n,m),m=1..n-1),n=2..10) ; # R. J. Mathar, Mar 04 2018

Table[Sum[StirlingS2[m + 1, i + 1] (-1)^(m - i) i^(n - m) i!, {i, 0, m} ], {n, 11}, {m, n - 1}] /. {} -> {0} // Flatten  (* Michael De Vlieger, May 19 2016 *)

A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
concat(vector(10, n, vector(n, m, A(n+1, m))))  \\ Gheorghe Coserea, May 16 2016

T(n,m) = Sum_{i=0..m} Stirling2(m+1, i+1)*(-1)^(m-i)*i^(n-m)*i!, for n>=2, m=1..n-1, where Stirling2(n,k) is defined by A008277.

A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - Gheorghe Coserea, May 18 2016

More terms from Gheorghe Coserea, May 16 2016

A216332 Number of horizontal and antidiagonal neighbor colorings of the even squares of an n X 2 array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 2, 3, 10, 27, 114, 409, 2066, 9089, 52922, 272947, 1788850, 10515147, 76282138, 501178937, 3974779402, 28773452321, 247083681522, 1949230218691, 17984917069018, 153281759047387, 1510073008031682, 13806215066685433

Offset: 1

R. H. Hardin, Sep 04 2012

nonn

Number of vertex covers and independent vertex sets of the n-1 X n-1 black bishops graph. Equivalently, the number of ways to place any number of non-attacking bishops on the black squares of an n-1 X n-1 board. - Andrew Howroyd, May 08 2017

			Some solutions for n=5:
..0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x....0..x
..x..1....x..1....x..1....x..0....x..1....x..1....x..0....x..1....x..1....x..0
..0..x....2..x....2..x....1..x....2..x....2..x....1..x....2..x....0..x....1..x
..x..2....x..0....x..1....x..2....x..1....x..0....x..1....x..0....x..1....x..2
..3..x....3..x....3..x....0..x....2..x....1..x....0..x....2..x....0..x....3..x
There are 5 black squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 5 ways to place 1 and 4 ways to place 2 so a(4)=1+5+4=10. - _Andrew Howroyd_, Jun 06 2017

R. H. Hardin, Table of n, a(n) for n = 1..210
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover

Column 2 of A216338.
Row sums of A274105(n-1) for n>2.
Cf. A216078, A201862, A286422.

Table[Sum[Binomial[Ceiling[n/2], k] BellB[n - k], {k, 0, Ceiling[n/2]}], {n, 0, 20}] (* Eric W. Weisstein, Jun 25 2017 *)

A288183 Triangle read by rows: T(n,k) = number of arrangements of k non-attacking bishops on the black squares of an n X n board with every square controlled by at least one bishop.

Original entry on oeis.org

2, 1, 4, 0, 4, 4, 0, 0, 22, 8, 0, 0, 16, 64, 8, 0, 0, 6, 128, 228, 16, 0, 0, 0, 72, 784, 528, 16, 0, 0, 0, 0, 1056, 4352, 1688, 32, 0, 0, 0, 0, 432, 9072, 18336, 3584, 32, 0, 0, 0, 0, 120, 7776, 76488, 87168, 11024, 64, 0, 0, 0, 0, 0, 2880, 109152, 484416, 313856, 22592, 64

Offset: 2

Andrew Howroyd, Jun 06 2017

See A146304 for algorithm and PARI code to produce this sequence.
Equivalently, the coefficients of the maximal independent set polynomials on the n X n black bishop graph.
The product of the first nonzero term in each row of this sequence and that of A288182 give A122749.

			Triangle begins:
  2;
  1, 4;
  0, 4,  4;
  0, 0, 22,   8;
  0, 0, 16,  64,    8;
  0, 0,  6, 128,  228,   16;
  0, 0,  0,  72,  784,  528,    16;
  0, 0,  0,   0, 1056, 4352,  1688,    32;
  0, 0,  0,   0,  432, 9072, 18336,  3584,    32;
  0, 0,  0,   0,  120, 7776, 76488, 87168, 11024,  64;
  ...
The first term is T(2,1) = 2.

Andrew Howroyd, Table of n, a(n) for n = 2..1276
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Row sums are A290594.

Cf. A288182, A122749, A274105, A146304.

A088960 Triangle read by rows: T(n,k) = number of configurations of k non-attacking bishops on the white squares of an n X n chessboard (for n even, 0 <= k < n).

Original entry on oeis.org

1, 2, 1, 8, 14, 4, 1, 18, 98, 184, 100, 8, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 72, 2050, 29900, 242292, 1109184, 2800016, 3653280, 2180656, 474368, 23264, 64

Offset: 2

Brant Jones (brant(AT)math.washington.edu), Oct 28 2003

			T(4,1) = 8 because there are 8 white squares on the 4 X 4 board to put one bishop; T(4,3) = 4 because we must place one bishop on each of three principal diagonal lines, which can be accomplished in 2*1*2=4 ways.
Triangle begins:
1, 2
1, 8, 14, 4
1, 18, 98, 184, 100, 8
1, 32, 356, 1704, 3532, 2816, 632, 16

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997; see section 2.4.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
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]
S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169, 2013

T[n_, k_] := (Sum[(-1)^j*Binomial[n - k - 1, j]/(n - k - 1)!*(n - k + 1 - j)^(n/2)*(n - k - j)^(n/2 - 1), {j, 0, n - k - 1}]); Flatten[Table[T[n, k], {n, 2, 12, 2}, {k, 0, n - 1}]] (* Vaclav Kotesovec, Mar 24 2011 *)

Generating function for fixed n = rook polynomial of Ferrers board with shape (2, 2, 4, 4, 6, 6, 8, 8, ..., (n-2), (n-2), n)

A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 1, 8, 14, 4, 1, 12, 38, 32, 4, 1, 18, 98, 184, 100, 8, 1, 24, 188, 576, 652, 208, 8, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 40, 580, 3840, 12052, 16944, 9080, 1280, 16, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32

Offset: 0

N. J. A. Sloane, Jun 14 2016

From Eder G. Santos, Dec 16 2024: (Start)
The sequence counts every possible nonattacking configuration of k bishops on the white squares of an n X n chess board.
It is assumed that the n X n chess board has a black square in the upper left corner.
(End)

			Triangle begins:
  1;
  1;
  1,  2;
  1,  4,    2;
  1,  8,   14,     4;
  1, 12,   38,    32,     4;
  1, 18,   98,   184,   100,      8;
  1, 24,  188,   576,   652,    208,      8;
  1, 32,  356,  1704,  3532,   2816,    632,     16;
  1, 40,  580,  3840, 12052,  16944,   9080,   1280,     16;
  1, 50,  940,  8480, 38932,  89256,  93800,  37600,   3856,   32;
  1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32;
  ...
From _Eder G. Santos_, Dec 16 2024: (Start)
For example, for n = 3, k = 2, the T(3,2) = 2 nonattacking configurations are:
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
  |   |   |   | , | B |   | B |
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
(End)

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]
J. Perott, Sur le problème des fous, Bulletin de la S. M. F., tome 11 (1883), pp. 173-186.
Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas. arXiv:2411.16492 [math.CO], 2024. (considered as black board).
Eric Weisstein's World of Mathematics, White Bishop Graph.

Columns k=0-1 give: A000012, A007590.
Alternate rows give A088960.
Row sums are A216078(n+1).
T(2n,n) gives A191236.
T(2n+1,n) gives A217900(n+1).
T(n+1,n) gives A060546.
Cf. A274105 (black squares), A288182, A201862, A002465.

with(combinat): with(gfun):
T := n -> add(stirling2(n+1,n+1-k)*x^k, k=0..n):
# bishops on white squares
bish := proc(n) local m,k,i,j,t1,t2; global T;
    if n=0 then return [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,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:

T[n_] := Sum[StirlingS2[n+1, n+1-k]*x^k, {k, 0, n}];
bish[n_] := Module[{m, t1, t2}, If[Mod[n, 2] == 0,
   m = n/2;     t1 = Sum[Binomial[m, k]*T[2*m-1-k]*x^k, {k, 0, m}],
   m = (n-1)/2; t1 = Sum[Binomial[m, k]*T[2*m - k]*x^k, {k, 0, m+1}]];
CoefficientList[t1 + O[x]^(2*n+1), x]];
Table[bish[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 25 2022, after Maple code *)

def stirling2_negativek(n, k):
  if k < 0: return 0
  else: return stirling_number2(n, k)
print([sum([binomial(floor(n/2), j)*stirling2_negativek(n-j, n-k) for j in [0..k]]) for n in [0..10] for k in [0..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(floor(n/2),j) * Stirling2(n-j,n-k).
T(n,k) = T(n-1,k) + (n-k+1-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

A297195 Number of bitriangular permutations (row sums of A272644 if that triangle is prefixed with two rows for n=0,1).

Original entry on oeis.org

1, 0, 1, 2, 7, 28, 133, 726, 4483, 30896, 235105, 1957930, 17712799, 172980804, 1813760317, 20323234814, 242353047355, 3064550705752, 40958281206169, 576917769130578, 8541793624670551, 132623408805525740, 2154730841214003061, 36560670776303600422, 646697046042017004787

Offset: 0

N. J. A. Sloane, Jan 10 2018

Define c(n) = Sum_{m=2..n-1} C(n-1, m-1)^(-2). Define b(1) = x and b(n+1) = b(n) + (Sum_{m=2..n-1} b(m)*b(n+1-m)*C(n-1, m-1)^(-2))/n^2 for n>0. Then b(n) is a polynomial in x and so is (b(n+1)-b(n))/x^2 whose constant term is c(n)/n^2. The Hone et.al.[2002] link denotes x with alpha_2 and alpha_k = (k-1)!^2*b(k). Conjecture: Asymptotic expansion of c(n) = 2*Sum_{i>1} a(i)/n^i. - Michael Somos, Oct 17 2024

			G.f. = 1 + x^2 + 2*x^3 + 7*x^4 + 28*x^5 + 133*x^6 + 726*x^7 + ... - _Michael Somos_, Oct 17 2024

A.N.W. Hone, N. Joshi and A.V. Kitaev, An Entire Function Defined by a Nonlinear Recurrence Relation, J. of the London Math. Soc., Oct. 2002, vol. 66, iss. 2, pp. 377-387.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]

Cf. A272644.

A297195 := proc(n)
    add(A272644(n, m), m=0..n) ;
end proc:
seq(A297195(n), n=0..30) ; # R. J. Mathar, Mar 04 2018

A272644[n_, m_] := Sum[StirlingS2[m+1, i+1] (-1)^(m-i) i^(n-m) i!, {i, 0, m}];
a[n_] := If[n == 1, 1, Sum[A272644[n, m], {m, 1, n-1}]];
Array[a, 24] (* Jean-François Alcover, Apr 03 2020 *)

{a(n) = if(n<2, n==0, sum(m=1, n-1, sum(i=0, m, (-1)^(m-i)*i^(n-m)*i!*stirling(m+1, i+1, 2))))}; /* Michael Somos, Oct 17 2024 */

Some terms corrected by Alois P. Heinz, Oct 17 2024

cp's OEIS Frontend

A036563 a(n) = 2^n - 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!, for n >= 2, m = 1..n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A216332 Number of horizontal and antidiagonal neighbor colorings of the even squares of an n X 2 array with new integer colors introduced in row major order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A288183 Triangle read by rows: T(n,k) = number of arrangements of k non-attacking bishops on the black squares of an n X n board with every square controlled by at least one bishop.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A088960 Triangle read by rows: T(n,k) = number of configurations of k non-attacking bishops on the white squares of an n X n chessboard (for n even, 0 <= k < n).

Views

Author

Keywords

Examples

References

Links

Programs

Mathematica

Formula

A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

Extensions

A297195 Number of bitriangular permutations (row sums of A272644 if that triangle is prefixed with two rows for n=0,1).

Views

Author

Keywords

Comments

Examples

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.