A222715 -id:A222715 - cp's OEIS frontend

A226048 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (2,n)-rectangular grid with k '1's and (2n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 6, 6, 6, 2, 1, 1, 2, 10, 14, 22, 14, 10, 2, 1, 1, 3, 15, 32, 60, 66, 60, 32, 15, 3, 1, 1, 3, 21, 55, 135, 198, 246, 198, 135, 55, 21, 3, 1, 1, 4, 28, 94, 266, 508, 777, 868, 777, 508, 266, 94, 28, 4, 1, 1, 4, 36

Offset: 0

Yosu Yurramendi, May 24 2013

Sum of rows (see example) gives A225826.
This triangle is to A225826 as Losanitsch's triangle A034851 is to A005418.
By columns:
T(n,1) is A004526.
T(n,2) is A000217.
T(n,3) is A225972.
T(n,4) is A071239.
T(n,5) is A222715.
T(n,6) is A228581.
T(n,7) is A228582.
T(n,8) is A228583.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014

			n\k 0 1  2   3   4   5   6   7   8   9  10 11 12 13 14
0   1
1   1 1  1
2   1 1  3   1   1
3   1 2  6   6   6   2   1
4   1 2 10  14  22  14  10   2   1
5   1 3 15  32  60  66  60  32  15   3   1
6   1 3 21  55 135 198 246 198 135  55  21  3  1
7   1 4 28  94 266 508 777 868 777 508 266 94 28  4  1
8   1 4 36 140...
...
The length of row n is 2*n+1, so n>= floor((k+1)/2).

Yosu Yurramendi and María Merino, Rows 0..40 of triangle, flattened

Cf. A225826, A005418, A034851.

A226048 := proc(n,k)
    if type(k,'even') then
        binomial(2*n,k) +3*binomial(n,k/2) ;
    else
        binomial(2*n,k) +(1-(-1)^n)*binomial(n-1,(k-1)/2) ;
    end if ;
    %/4 ;
end proc:
seq(seq(A226048(n,k),k=0..2*n),n=0..8) ; # R. J. Mathar, Jun 07 2020

T[n_, k_] := If[EvenQ[k],
   Binomial[2n, k] + 3 Binomial[n, k/2],
   Binomial[2n, k] + (1-(-1)^n) Binomial[n-1, (k-1)/2]]/4;
Table[T[n, k], {n, 0, 8}, { k, 0, 2n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

If k even, 4*T(n,k) = binomial(2*n,k) +3*binomial(n,k/2). - Yosu Yurramendi, María Merino, Aug 25 2013

If k odd, 4*T(n,k) = 4*T(n,k) = binomial(2*n,k) +(1-(-1)^n)*binomial(n-1,(k-1)/2). - Yosu Yurramendi, María Merino, Aug 25 2013 [corrected by Christian Barrientos, Jun 14 2018]

Definition corrected by María Merino, May 19 2017

A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 14, 39, 14, 1, 3, 66, 208, 208, 66, 3, 12, 198, 794, 1092, 794, 198, 12, 28, 508, 2196, 3912, 3912, 2196, 508, 28, 66, 1092, 5231, 10626, 13462, 10626, 5231, 1092, 66, 126, 2156, 10808, 24648, 35787, 35787, 24648, 10808, 2156, 126

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

			T(n,k) for 1<=n<=8 and 1<=k<=8 is:
.  k   1      2      3      4      5      6      7       8 ...
n
1      0      0      0      0      1      3     12      28
2      0      0      2     14     66    198    508    1092
3      0      2     39    208    794   2196   5231   10808
4      0     14    208   1092   3912  10626  24648   50344
5      1     66    794   3912  13462  35787  81648  164980
6      3    198   2196  10626  35787  94248 212988  428076
7     12    508   5231  24648  81648 212988 477903  955856
8     28   1092  10808  50344 164980 428076 955856 1906128

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n
   + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4
   - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3
   - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2
   + 48*k + 48*n + 45
   + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3
      + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n
      - 45)*(-1)^k
   + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n
      - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n
      - 48*k - 45)*(-1)^n
   + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4 - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3 - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2 + 48*k + 48*n + 45
+ (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3 + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n - 45)*(-1)^k
+ (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n - 48*k - 45)*(-1)^n
+ (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
T(1,k) = A005995(k-5) = (k-3)*(k-1)*((k-4)*(k-2)*2*k + 15*(1-(-1)^k))/480;
T(2,k) = A222715(k) = (k-2)*(k-1)*((2*k-3)(2*k-1)*2*k + 15*(1-(-1)^k))/120.

Terms corrected and extended by Christopher Hunt Gribble, Apr 16 2015

A241262 Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 10, 6, 14, 42, 28, 10, 42, 198, 165, 60, 15, 132, 1001, 1092, 455, 110, 21, 429, 5304, 7752, 3876, 1020, 182, 28, 1430, 29070, 57684, 35420, 10626, 1995, 280, 36, 4862, 163438, 444015, 339300, 118755, 24570, 3542, 408, 45, 16796, 937365, 3506100, 3362260, 1391280, 324632, 50344, 5850, 570, 55

Offset: 1

Jean-François Alcover, Apr 18 2014

About the "root estimation" question asked in MathOverflow, one can check (at least numerically) that, for instance with k = 4 and a = 1/11, the series a^-1 + (k - 1) + Sum_{n>=} (-1)^n*binomial(n*k, n+1)/n*a^n evaluates to the positive solution of x^k = (x+1)^(k-1).
Row 1 is A000217 (triangular numbers),
Row 2 is A006331 (twice the square pyramidal numbers),
Row 3 is A067047(3n) = lcm(3n, 3n+1, 3n+2, 3n+3)/12 (from column r=4 of A067049),
Row 4 is A222715(2n) = (n-1)*n*(2n-1)*(4n-3)*(4n-1)/15,
Row 5 is not in the OEIS.
Column 1 is A000108 (Catalan numbers),
Column 2 is A007226 left shifted 1 place,
Column 4 is A007228 left shifted 1 place,
Column 5 is A124724 left shifted 1 place,
Column 6 is not in the OEIS.

			Array begins:
    1,    3,     6,     10,      15,      21, ...
    2,   10,    28,     60,     110,     182, ...
    5,   42,   165,    455,    1020,    1995, ...
   14,  198,  1092,   3876,   10626,   24570, ...
   42, 1001,  7752,  35420,  118755,  324632, ...
  132, 5304, 57684, 339300, 1391280, 4496388, ...
  etc.

N. S. S. Gu, H. Prodinger, S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat. 31 (2010) 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2

MathOverflow, Root estimation

Cf. A000217, A006331, A000108, A007226, A007228, A067047, A067049, A124724, A222715.

t[n_, k_] := Binomial[n*k, n+1]/n; Table[t[n-k+2, k], {n, 1, 10}, {k, 2, n+1}] // Flatten

cp's OEIS Frontend

A226048 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (2,n)-rectangular grid with k '1's and (2n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

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Programs

Maple

Formula

Extensions

A241262 Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

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