A008805 -id:A008805 - cp's OEIS frontend

A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 6, 3, 4, 3, 1, 0, 6, 6, 6, 4, 1, 10, 6, 9, 10, 9, 5, 1, 0, 10, 12, 15, 16, 13, 6, 1, 15, 10, 16, 21, 25, 25, 18, 7, 1, 0, 15, 20, 28, 36, 41, 38, 24, 8, 1, 21, 15, 25, 36, 49, 61, 66, 56, 31, 9, 1, 0, 21, 30, 45, 64, 85, 102, 104, 80, 39, 10, 1, 28, 21, 36, 55, 81, 113, 146, 168, 160, 111, 48, 11, 1

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n, 0) = T(n, 3) with T(2n, 0) = T(n, m) for some other value of m would change the generating function to the coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^m. This would produce A058393, A058394, A057884 (and effectively A007318).

			The array T(n, k) starts:
[0] 1, 0,  3,   0,   6,   0,  10,    0,   15,    0, ...
[1] 1, 1,  3,   3,   6,   6,  10,   10,   15,   15, ...
[2] 1, 2,  4,   6,   9,  12,  16,   20,   25,   30, ...
[3] 1, 3,  6,  10,  15,  21,  28,   36,   45,   55, ...
[4] 1, 4,  9,  16,  25,  36,  49,   64,   81,  100, ...
[5] 1, 5, 13,  25,  41,  61,  85,  113,  145,  181, ...
[6] 1, 6, 18,  38,  66, 102, 146,  198,  258,  326, ...
[7] 1, 7, 24,  56, 104, 168, 248,  344,  456,  584, ...
[8] 1, 8, 31,  80, 160, 272, 416,  592,  800, 1040, ...
[9] 1, 9, 39, 111, 240, 432, 688, 1008, 1392, 1840, ...

Rows are A000217 with zeros, A008805, A002620, A000217, A000290, A001844, A005899.
Columns are A000012, A001477, A016028.
Diagonals include A058396, A049611, A001793, A001788, A055580, A055581, A055582.
The triangle A055252 also appears in half of the array.

gf := n -> (1 + x)^n / (1 - x^2)^3: ser := n -> series(gf(n), x, 20):
seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..9)), n = 0..9); # Peter Luschny, Apr 12 2023

T[0, k_] := If[OddQ[k], 0, (k+2)(k+4)/8];
T[n_, k_] := T[n, k] = If[k == 0, 1, T[n-1, k-1] + T[n-1, k]];
Table[T[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 13 2023 *)

T(n, k) = T(n-1, k-1) + T(n, k-1) with T(0, k) = 1, T(2*n, 0) = T(n, 3) and T(2*n + 1, 0) = 0. Coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^3.

A189980 a(n) is the number of incongruent two-color bracelets of n beads, 10 from them are black (A005515), having a diameter of symmetry.

Original entry on oeis.org

1, 1, 6, 6, 21, 21, 56, 56, 126, 126, 252, 252, 462, 462, 792, 792, 1287, 1287, 2002, 2002, 3003, 3003, 4368, 4368, 6188, 6188, 8568, 8568, 11628, 11628, 15504, 15504, 20349, 20349, 26334, 26334, 33649, 33649

Offset: 10

Vladimir Shevelev, May 03 2011

For n >= 11, a(n-1) is the number of incongruent two-color bracelets of n beads, 11 from them are black (A032282), having a diameter of symmetry.

Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Cf. A005515, A032282, A008805, A058187, A189976.

A189980 :=proc(n): binomial(floor(n/2),5) end: seq(A189980(n), n=10..47); # Johannes W. Meijer, Aug 15 2011

Table[Binomial[Floor[n/2],5],{n,10,50}] (* Harvey P. Dale, Oct 06 2017 *)

a(n) = binomial(floor(n/2), 5). [Typo fixed by Colin Barker, Feb 07 2013]
a(n+6) = A194005(n, n-5). - Johannes W. Meijer, Aug 15 2011
G.f.: x^10/((x-1)^6*(x+1)^5). - Colin Barker, Feb 07 2013

Data added and link corrected by Johannes W. Meijer, Aug 15 2011

A242279 Number of inequivalent (mod D_4) ways four checkers can be placed on an n X n board.

Original entry on oeis.org

1, 23, 252, 1666, 7509, 26865, 79920, 209096, 491425, 1064575, 2150076, 4104738, 7458437, 13005041, 21857984, 35598880, 56353185, 87019191, 131364700, 194364050, 282314901, 403316353, 567402672, 787201416, 1078078209, 1459020095, 1952782300, 2587048786, 3394568325

Offset: 2

Heinrich Ludwig, May 10 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

Cf. A054252, A008805, A014409, A082966, A242358, A019318, A054772.

CoefficientList[Series[x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5), {x, 0, 20}], x] (* Vaclav Kotesovec, May 10 2014 *)
LinearRecurrence[{4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1},{0,0,1,23,252,1666,7509,26865,79920,209096,491425,1064575,2150076,4104738},40] (* Harvey P. Dale, May 06 2018 *)

a(n) = (n^8 - 6*n^6 + 40*n^4 - 48*n^3 + 16*n^2 + IF(MOD(n, 2) = 1)*(14*n^4 - 48*n^3 + 34*n^2 - 3))/192.
G.f.: x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5). - Vaclav Kotesovec, May 10 2014
a(n) = A054772(n, 4), n >= 2. - Wolfdieter Lang, Oct 03 2016

A242358 Number of inequivalent (mod D_4) ways five checkers can be placed on an n X n board.

Original entry on oeis.org

23, 567, 6814, 47358, 239511, 954226, 3207212, 9414828, 24862239, 60136329, 135311658, 286229762, 574460495, 1101240084, 2028333848, 3605765688, 6211552455, 10402472811, 16984387958, 27099325638, 42342870823, 64905898662, 97761436356, 144885584740, 211543443215

Offset: 3

Heinrich Ludwig, May 11 2014

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A054252, A008805, A014409, A082966, A242279, A054772.

Drop[CoefficientList[Series[x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6), {x, 0, 20}], x],3] (* Vaclav Kotesovec, May 11 2014 *)

a(n) = (n^10 - 10*n^8 + 35*n^6 + 52*n^5 - 210*n^4 + 140*n^3 - 56*n^2 + 48*n + IF(MOD(n, 2) = 1)*(52*n^5 - 145*n^4 + 140*n^3 - 80*n^2 + 48*n - 15))/960.
G.f.: x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6). - Vaclav Kotesovec, May 11 2014
a(n) = A054772(n, 5), n >=3. - Wolfdieter Lang, Oct 03 2016

A330641 a(n) is the number of partitions of n with Durfee square of size <= 3.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 230, 295, 380, 480, 607, 758, 943, 1161, 1426, 1733, 2100, 2525, 3023, 3595, 4261, 5017, 5888, 6874, 7996, 9258, 10687, 12281, 14073, 16066, 18288, 20747, 23478, 26482, 29801, 33442, 37441, 41811, 46596, 51801, 57478, 63639, 70329, 77567

Offset: 0

Omar E. Pol, Dec 22 2019

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A330640.

seq(n)={Vec((1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - x^7 + x^8 + x^9 - x^11 + x^12) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x*x^n))} \\ Andrew Howroyd, Dec 27 2024

a(n) = A000041(n), 0 <= n <= 15.
a(n) = A330640(n), 0 <= n <= 8.
a(n) = A330640(n) + A117485(n), n >= 9.
a(n) = n + A006918(n-3) + A117485(n), n >= 9.
From Colin Barker, Dec 31 2019: (Start)
G.f.: (1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - x^7 + x^8 + x^9 - x^11 + x^12) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>12. (End)

A064349 Generating function: 1/((1-x)(1-x^2)^2(1-x^3)^3*(1-x^4)^4).

Original entry on oeis.org

1, 1, 3, 6, 13, 19, 37, 58, 97, 143, 227, 328, 492, 688, 992, 1364, 1903, 2551, 3473, 4586, 6097, 7911, 10333, 13226, 16988, 21454, 27172, 33938, 42437, 52423, 64833, 79354, 97130, 117824, 142930, 172018, 206925, 247179, 295105, 350154, 415124

Offset: 0

Henry Bottomley, Sep 17 2001

nonn

Number of partitions of n into parts 1 (of one kind), 2 (of two kinds), 3 (of three kinds), and 4 (of 4 kinds). [Joerg Arndt, Jul 11 2013]

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 2, 1, 0, -9, -5, 2, 13, 21, -4, -17, -30, -13, 25, 28, 25, -13, -30, -17, -4, 21, 13, 2, -5, -9, 0, 1, 2, 1, -1).

The sequence of sequences A000007, A000012, A008805, A002597, A064349, etc. approaches A000219.
Essentially the same as A002598.
Cf. A002598.

a(n) = floor( ([13, 28, -44][n%3+1]+(9/2)*(n\3+2)*((n+1)%3-1)) * (n\3+1)/729 - (n\2+1)*(-1)^(n\2) * (3*[-8, 11]+(n\2+2)*(2*[-1, 3]+(n\2+3)*(1/3)*[0, 1]))[n%2+1]/512 + (2*n^9 +270*n^8 +15600*n^7 +504000*n^6 +9977730*n^5 +124629750*n^4 +973069200*n^3 +4521339000*n^2 +11137512613*n +16461579435 +5103*(n+15)*(2*n^4 +120*n^3 +2440*n^2 +19200*n +48213)*(-1)^n) / 20065812480 ) \\ Tani Akinari, Jul 12 2013

Vec(1/((1-x)*(1-x^2)^2*(1-x^3)^3*(1-x^4)^4)+O(x^66)) \\ Joerg Arndt, Jul 11 2013

A157898 Triangle read by rows: inverse binomial transform of A059576.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 2, 2, 4, 1, 2, 6, 4, 8, 0, 3, 6, 16, 8, 16, 1, 3, 12, 16, 40, 16, 32, 0, 4, 12, 40, 40, 96, 32, 64, 1, 4, 20, 40, 120, 96, 224, 64, 128, 0, 5, 20, 80, 120, 336, 224, 512, 128, 256

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 08 2009

The inverse binomial transform of the triangle A059576 is given by multiplying the triangle with A130595 from the left.

			First few rows of the triangle =
  1;
  0, 1;
  1, 1,  2;
  0, 2,  2,  4;
  1, 2,  6,  4,   8;
  0, 3,  6, 16,   8,  16;
  1, 3, 12, 16,  40,  16,  32;
  0, 4, 12, 40,  40,  96,  32,  64;
  1, 4, 20, 40, 120,  96, 224,  64, 128;
  0, 5, 20, 80, 120, 336, 224, 512, 128, 256;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A059576, A097076 (row sums), A130595.

Cf. A004526, A008805, A011782, A058187, A059841, A189976.

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*t(j,k): j in [k..n]]) >;
[A157898(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 03 2022

A059576 := proc (n, k)
    if n = 0 then
        return 1;
    end if;
    if k <= n and k >= 0 then
        add((-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k), j = 0 .. min(k, n-k))
    else
        0 ;
    end if
end proc:
A157898 := proc(n,k)
    add ( A130595(n,j)*A059576(j,k),j=k..n) ;
end proc: # R. J. Mathar, Feb 13 2013

t[n_, k_]:= t[n, k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1, k-1] + 2*t[n-1, k] -(2 -Boole[n==2])*t[n-2, k-1]]; (* t= A059576 *)
A157898[n_, k_]:= A157898[n, k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
Table[A157898[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 03 2022 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n,k): return sum((-1)^(n-j)*binomial(n,j)*t(j,k) for j in (k..n))
flatten([[A157898(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 03 2022

Sum_{k=0..n} T(n, k) = A097076(n+1).
From G. C. Greubel, Sep 03 2022: (Start)
T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(n,j)*A059576(j,k).
T(n, 0) = A059841(n).
T(n, 1) = A004526(n-1).
T(n, 2) = 2*A008805(n-2).
T(n, 3) = 4*A058187(n-3).
T(n, 4) = 8*A189976(n+4).
T(n, n) = A011782(n).
T(n, n-1) = A011782(n) - [n==0]. (End)

A211534 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w = 3x + 3y.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 3, 3, 6, 6, 6, 10, 10, 10, 15, 15, 15, 21, 21, 21, 28, 28, 28, 36, 36, 36, 45, 45, 45, 55, 55, 55, 66, 66, 66, 78, 78, 78, 91, 91, 91, 105, 105, 105, 120, 120, 120, 136, 136, 136, 153, 153, 153, 171, 171, 171, 190, 190, 190, 210

Offset: 0

Clark Kimberling, Apr 15 2012

This sequence consists of six 0's followed by triply repeated triangular numbers.

For a guide to related sequences, see A211422.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A211422, A008805 (w = 2x + 2y and doubly repeated triangular numbers).

[Floor(n/3)*(Floor(n/3)-1)/2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 05 2015

[n le 7 select Floor(n/7) else Self(n-1)+2*Self(n-3)-2*Self(n-4)-Self(n-6)+ Self(n-7): n in [1..70]]; // Vincenzo Librandi, Apr 05 2015

A211534:=n->floor(n/3)*(floor(n/3)-1)/2: seq(A211534(n), n=0..100); # Wesley Ivan Hurt, Apr 05 2015

t[n_] := t[n] = Flatten[Table[-w + 3 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211534 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {0, 0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Apr 05 2015 *)

concat([0,0,0,0,0,0], Vec(-x^6/((x-1)^3*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, Feb 17 2015

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).
a(n) = floor(n/3)*( floor(n/3) - 1 )/2. - Luce ETIENNE, Jul 08 2014
G.f.: -x^6 / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Feb 17 2015
a(n) = Sum_{i=0..n-3} i*0^(i mod 3)/3. - Wesley Ivan Hurt, Apr 05 2015

A247976 Triangle read by rows: T(n,k) generated by m-gon expansions in the case of odd m with "vertex to vertex" version or even m with "vertex to side" version. (See comment for details.)

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7

Offset: 1

Kival Ngaokrajang, Sep 28 2014

Refer to triangle expansions in A061777 and A101946 (and their companions for m-gons) which are "vertex to vertex" and "vertex to side" versions respectively. The label values at each iteration can be arranged as triangle. Any m-gon can also be arranged as the same triangle with conditions: (i) m is odd and expansion is "vertex to vertex" version or (ii) m is even and expansion is "vertex to side" version. m*Sum_{i=1..k}T(n,k) gives the total label value in n-th iteration. See illustration.

			Triangle begins:
  1;
  1,  1;
  1,  1,  2;
  1,  2,  1,  2;
  1,  2,  1,  3,  3;
  1,  3,  3,  1,  3,  3;
  1,  3,  3,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  5, 10, 10,  5;
  1,  5, 10, 10,  5,  1,  5, 10, 10, 5;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Kival Ngaokrajang, Illustration of initial terms

Rows sum: A027383.
Column (start from 1s): c3=A008805, c4=A058187, c5=A000332 repeated, c6=A000389 repeated, c7=A000579 repeated.
Vertex to vertex: A061777, A247618, A247619, A247620.
Vertex to side: A101946, A247903, A247904, A247905.
Cf. A074909.

T[n_, k_]:= T[n, k]= If[k==1, 1, If[k==n, Floor[(n+1)/2], If[OddQ[n], If[k<=(n+ 1)/2, T[n-1, k], T[n-1, k-1] + T[n-1, k]], If[kG. C. Greubel, Feb 18 2022 *)

@CachedFunction
def T(n,k): # A247976
    if (k==1): return 1
    elif (k==n): return (n+1)//2
    elif (n%2==1): return T(n-1,k) if (k <= (n+1)/2) else T(n-1,k-1) + T(n-1,k)
    else: return T(n-1,k-1)+T(n-1,k) if (k < (n+2)/2) else T(n,k-n/2)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Feb 18 2022

T(n, k) = ( T(n-1, k) if k <= (n+1)/2 otherwise T(n-1, k-1) + T(n-1, k) ) for odd n rows, ( T(n-1, k-1) + T(n-1, k) if k < (n+2)/2 otherwise T(n, k - n/2) ) for even n rows, with T(n, 1) = 1 and T(n, n) = floor((n+1)/2). - G. C. Greubel, Feb 18 2022

A317575 Irregular triangle read by rows: T(n,k) is the number of n X n tic-tac-toe positions (up to rotation and reflection) with k tokens (i.e., after k plays) which occur in a game (n > 0, 0 <= k <= n^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 0, 1, 3, 12, 38, 108, 174, 204, 153, 57, 15, 1, 3, 33, 219, 1413, 5514, 20122, 50215, 112379, 179510, 245690, 245690, 193318, 110452, 41870, 10489, 1059, 1, 6, 85, 904, 9664, 66859

Offset: 1

Álvar Ibeas, Jul 31 2018

			Triangle begins:
  n\k |  0   1   2   3   4   5   6   7   8   9
  ----+---------------------------------------
   1  |  1   1
   2  |  1   1   2   2   0
   3  |  1   3  12  38 108 174 204 153  57  15

Index entries for sequences related to tic-tac-toe

Cf. A008907 (3rd row), A008805 (column k=1), A242709 (column k=2), A317573, A317574.

cp's OEIS Frontend

A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

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A189980 a(n) is the number of incongruent two-color bracelets of n beads, 10 from them are black (A005515), having a diameter of symmetry.

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Mathematica

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A242279 Number of inequivalent (mod D_4) ways four checkers can be placed on an n X n board.

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Mathematica

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A242358 Number of inequivalent (mod D_4) ways five checkers can be placed on an n X n board.

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Mathematica

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A330641 a(n) is the number of partitions of n with Durfee square of size <= 3.

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PARI

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A064349 Generating function: 1/((1-x)*(1-x^2)^2*(1-x^3)^3*(1-x^4)^4).

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PARI

PARI

A157898 Triangle read by rows: inverse binomial transform of A059576.

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Magma

Maple

Mathematica

SageMath

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A211534 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w = 3x + 3y.

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A064349 Generating function: 1/((1-x)(1-x^2)^2(1-x^3)^3*(1-x^4)^4).