A006918 -id:A006918 - cp's OEIS frontend

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

1, 3, 9, 20, 42, 78, 139, 231, 372, 573, 861, 1254, 1791, 2499, 3432, 4629, 6162, 8085, 10492, 13455, 17094, 21503, 26832, 33201, 40795, 49764, 60333, 72687, 87096, 103785, 123075, 145236, 170646, 199626, 232617, 269997, 312277, 359898, 413448, 473438

Offset: 1

Alford Arnold, Aug 21 2009

Convolution of A006918 with A001399, or of A002625 with A059841 (A000035 if offsets are respected),
or of A038163 with A022003 or of A057524 with A027656 or of A014125 with the aerated version of A000217,
or of A002624 with A103221, or of A002623 with A008731, or of other combinations of splitting the signature -/3,3,1 into two components.
If we apply the enumeration of Molien series as described in A139672,
this is row 45=9*5 of a table of values related to Molien series, i.e., the
product of the sequence on row 9 (A006918) with the sequence on row 5 (A001399).
This is associated with the root system E6, and can be described using the additive function on the affine E6 diagram:
      1
      |
      2
      |
1--2--3--2--1

			To calculate a(3), we consider the first three terms of A001399 = (1 1 2...)
and the first three terms of A006918 = (1 2 5 ...), to get the convolved a(3) = 1*5+1*2+2*1 = 9.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Cf. A139672 (row 21).
For G2, the corresponding sequence is A001399.
For F4, the corresponding sequence is A115264.
For E7, the corresponding sequence is A210068.
For E8, the corresponding sequence is A045513.
See A210634 for a closely related sequence.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^3*(1-x^2)^3*(1-x^3)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(x/((1-x)^3*(1-x^2)^3*(1-x^3)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Jan 13 2020

Rest@CoefficientList[Series[x/((1-x)^3*(1-x^2)^3*(1-x^3)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{1,3,9,20,42,78,139,231,372,573,861,1254},40] (* Harvey P. Dale, Aug 03 2025 *)

Vec(1/(1-x)^3/(1-x^2)^3/(1-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

x=PowerSeriesRing(QQ, 'x', 40).gen()
1/((1-x)^3*(1-x^2)^3*(1-x^3))

a(n) = round( -(-1)^n*(n+3)*(n+7)/256 +(6*n^6 +180*n^5 +2070*n^4 +11400*n^3 +30429*n^2 +34290*n +9785)/103680 ) - R. J. Mathar, Mar 19 2012

Edited and extended by R. J. Mathar, Aug 22 2009

Corrected link to index entries - R. J. Mathar, Aug 26 2009

A325180 Heinz number of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

Original entry on oeis.org

5, 8, 10, 12, 20, 21, 35, 36, 42, 49, 54, 60, 63, 70, 81, 84, 90, 98, 100, 105, 126, 135, 140, 147, 150, 189, 196, 210, 225, 275, 294, 315, 385, 441, 500, 539, 550, 605, 700, 750, 770, 825, 847, 980, 1050, 1078, 1100, 1125, 1155, 1210, 1250, 1331, 1372, 1375

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The enumeration of these partitions by sum is given by A325182.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    5: {3}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   20: {1,1,3}
   21: {2,4}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   49: {4,4}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   81: {2,2,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  105: {2,3,4}

Gus Wiseman, Young diagrams corresponding to the first 96 terms.

Numbers k such that A263297(k) - A257990(k) = 2.
Positions of 2's in A325178.
Cf. A006918, A056239, A093641, A112798, A325164, A325170, A325179, A325182, A325192, A325197.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Select[Range[1000],codurf[#]-durf[#]==2&]

A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

Original entry on oeis.org

1, 2, 5, 4, 8, 10, 11, 8, 20, 16, 17, 20, 20, 22, 42, 16, 26, 40, 29, 32, 58, 34, 35, 40, 53, 40, 74, 44, 44, 84, 47, 32, 90, 52, 94, 80, 56, 58, 106, 64, 62, 116, 65, 68, 174, 70, 71, 80, 102, 106, 138, 80, 80, 148, 146, 88, 154, 88, 89, 168, 92, 94, 241, 64, 172

Offset: 1

Ilya Gutkovskiy, Oct 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A328203, Sequence Machine.

Cf. A000005, A000079 (fixed points), A000203, A001227, A002131, A003602, A006519, A006918, A026741, A038040, A109168, A113415, A140472, A152211, A193356, A245579, A349353 (Dirichlet inverse), A349354.

Cf. also A347957.

a:=[]; for k in [1..65] do if IsOdd(k) then a[k]:=(k * #Divisors(k) + DivisorSigma(1,k)) / 2; else a[k]:=(k * (#Divisors(k) - #Divisors(k div 2)) + DivisorSigma(1,k) - DivisorSigma(1,k div 2)) / 2;  end if; end for; a; // Marius A. Burtea, Oct 07 2019

nmax = 65; CoefficientList[Series[Sum[k x^k/(1 - x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, (n Mod[#, 2] + Boole[OddQ[n/#]] #)/2 &]; Table[a[n], {n, 1, 65}]

A328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2)); \\ Antti Karttunen, Nov 13 2021

a(n) = (n * d(n) + sigma(n)) / 2 if n odd, (n * (d(n) - d(n/2)) + sigma(n) - sigma(n/2)) / 2 if n even.
a(n) = (n * A001227(n) + A002131(n)) / 2.
a(2*n) = 2 * a(n).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. Both are easy to prove:
a(n) = Sum_{d|n} A003602(d) * A026741(n/d).
a(n) = Sum_{d|n} A109168(d) * A193356(n/d), where A109168(d) = A140472(d) = (d+A006519(d))/2.
(End)

A059594 Convolution triangle based on A008619 (positive integers repeated).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 5, 3, 1, 3, 8, 9, 4, 1, 3, 14, 19, 14, 5, 1, 4, 20, 39, 36, 20, 6, 1, 4, 30, 69, 85, 60, 27, 7, 1, 5, 40, 119, 176, 160, 92, 35, 8, 1, 5, 55, 189, 344, 376, 273, 133, 44, 9, 1, 6, 70, 294, 624, 820, 714, 434

Offset: 0

Wolfdieter Lang, Feb 02 2001

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is 1/((1-z^2)*(1-z)-x*z).
The column sequences are A008619(n); A006918(n); A038163(n-2), n >= 2; A038164(n-3), n >= 3; A038165(n-4), n >= 4; A038166(n-5), n >= 5; A059595(n-6), n >= 6; A059596(n-7), n >= 7; A059597(n-8), n >= 8; A059598(n-9), n >= 9; A059625(n-10), n >= 10 for m=0..10.
The sequence of row sums is A006054(n+2).
From Gary W. Adamson, Aug 14 2016: (Start)
The sequence can be generated by extracting the descending antidiagonals of an array formed by taking powers of the natural integers with repeats, (1, 1, 2, 2, 3, 3, ...), as follows:
  1, 1,  2,  2,  3,   3, ...
  1, 2,  5,  8, 14,  20, ...
  1, 3,  9, 19, 39,  69, ...
  1, 4, 14, 36, 85, 176, ...
  ...
Row sums of the triangle = (1, 2, 5, 11, 25, 56, ...), the INVERT transform of (1, 1, 2, 2, 3, 3, ...). (End)

			{1}; {1,1}; {2,2,1}; {2,5,3,1}; ...
Fourth row polynomial (n=3): p(3,x)= 2 + 5*x + 3*x^2 + x^3.

t[n_, m_] := Sum[Sum[Binomial[j, n-m-3*k+2*j]*(-1)^(j-k)*Binomial[k, j], {j, 0, k}]*Binomial[m+k, m], {k, 0, n-m}]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, May 27 2013, after Vladimir Kruchinin *)

T(n,m):=sum((sum(binomial(j,n-m-3*k+2*j)*(-1)^(j-k)*binomial(k,j),j,0,k)) *binomial(m+k,m),k,0,n-m); /* Vladimir Kruchinin, Dec 14 2011 */

a(n, m) := a(n-1, m) + (-(n-m+1)*a(n, m-1) + 3*(n+2*m)*a(n-1, m-1))/(8*m), n >= m >= 1; a(n, 0) := floor((n+2)/2) = A008619(n), n >= 0; a(n, m) := 0 if n < m.
G.f.for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/((1-x^2)*(1-x)).
T(n,m) = Sum_{k=0..n-m} (Sum_{j=0..k} binomial(j, n-m-3*k+2*j)*(-1)^(j-k)*binomial(k,j))*binomial(m+k,m). - Vladimir Kruchinin, Dec 14 2011
Recurrence: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-3,k) with T(0,0) = 1. - Philippe Deléham, Feb 23 2012

A243866 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing one 1 X 1 tile in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 3, 3, 4, 4, 3, 3, 4, 3, 6, 4, 6, 3, 4, 4, 4, 6, 6, 6, 6, 4, 4, 5, 4, 8, 6, 9, 6, 8, 4, 5, 5, 5, 8, 8, 9, 9, 8, 8, 5, 5, 6, 5, 10, 8, 12, 9, 12, 8, 10, 5, 6, 6, 6, 10, 10, 12, 12, 12, 12, 10, 10, 6, 6, 7, 6, 12, 10, 15

Offset: 1

Christopher Hunt Gribble, Jun 19 2014

It appears that the number of equivalence classes of ways of placing one m X m tile in an n X k rectangle under all symmetry operations of the rectangle is T(n-m+1,k-m+1) for m >= 2, n >= m and k >= m, and zero otherwise. - Christopher Hunt Gribble, Oct 17 2014
The sum over each antidiagonal of A243866
= Sum_{j=1..n}(2*j + 1 - (-1)^j)*(2*(n - j + 1) + 1 - (-1)^(n - j + 1))/16
= (n + 2)*(2*n^2 + 8*n + 3 - 3*(-1)^n)/48
- see A006918. - Christopher Hunt Gribble, Apr 01 2015

			T(n,k) for 1<=n<=11 and 1<=k<=11 is:
    k    1    2    3    4    5    6    7    8    9   10   11 ...
.n
.1       1    1    2    2    3    3    4    4    5    5    6
.2       1    1    2    2    3    3    4    4    5    5    6
.3       2    2    4    4    6    6    8    8   10   10   12
.4       2    2    4    4    6    6    8    8   10   10   12
.5       3    3    6    6    9    9   12   12   15   15   18
.6       3    3    6    6    9    9   12   12   15   15   18
.7       4    4    8    8   12   12   16   16   20   20   24
.8       4    4    8    8   12   12   16   16   20   20   24
.9       5    5   10   10   15   15   20   20   25   25   30
10       5    5   10   10   15   15   20   20   25   25   30
11       6    6   12   12   18   18   24   24   30   30   36
...

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

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248017, A248027.

b := proc (n,k);
floor((1/2)*n+1/2)*floor((1/2)*k+1/2)
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = floor((n+1)/2)*floor((k+1)/2)
       = (2*n+1-(-1)^n)*(2*k+1-(-1)^k)/4;
T(n,1) = A034851(n,1);
T(n,2) = A226048(n,1);
T(n,3) = A226290(n,1);
T(n,4) = A225812(n,1);
T(n,5) = A228022(n,1);
T(n,6) = A228165(n,1);
T(n,7) = A228166(n,1). - Christopher Hunt Gribble, May 01 2015

Terms corrected by Christopher Hunt Gribble, Mar 27 2015

A244306 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing two 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 6, 6, 4, 6, 10, 13, 10, 6, 9, 15, 22, 22, 15, 9, 12, 21, 34, 36, 34, 21, 12, 16, 28, 48, 56, 56, 48, 28, 16, 20, 36, 65, 78, 88, 78, 65, 36, 20, 25, 45, 84, 106, 123, 123, 106, 84, 45, 25, 30, 55, 106, 136, 168, 171, 168, 136, 106, 55, 30

Offset: 1

Christopher Hunt Gribble, Jun 25 2014

			T(n,k) for 1<=n<=11 and 1<=k<=11 is:
    k  1    2    3    4    5    6    7    8    9   10   11 ...
.n
.1     0    1    2    4    6    9   12   16   20   25   30
.2     1    3    6   10   15   21   28   36   45   55   66
.3     2    6   13   22   34   48   65   84  106  130  157
.4     4   10   22   36   56   78  106  136  172  210  254
.5     6   15   34   56   88  123  168  216  274  335  406
.6     9   21   48   78  123  171  234  300  381  465  564
.7    12   28   65  106  168  234  321  412  524  640  777
.8    16   36   84  136  216  300  412  528  672  820  996
.9    20   45  106  172  274  381  524  672  856 1045 1270
10    25   55  130  210  335  465  640  820 1045 1275 1550
11    30   66  157  254  406  564  777  996 1270 1550 1885

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

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244307, A248011, A248016, A248059, A248060, A248017, A248027.

Empirically,
T(n,k) = (4*k^2*n^2 + 2*k^2 + 8*k*n + 2*n^2 - 4*k - 4*n - 1 - (2*k^2 - 4*k - 1)*(-1)^n - (2*n^2 - 4*n - 1)*(-1)^k - (-1)^k*(-1)^n)/32.
T(1,k) = A002620(k) = floor(k^2/4).
T(2,k) = A000217(k) = k*(k+1)/2.
       = T(1,k) + T(1,k+1) = floor(k^2/4) + floor((k+1)^2/4).
T(3,k) = 2*A000217(k) + A024206(k-2)
       = k*(k+1) + floor((k-1)^2/4) - 1.

Terms corrected and extended by Christopher Hunt Gribble, Apr 02 2015

A248011 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing three 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, 1, 1, 1, 2, 6, 6, 2, 6, 14, 27, 14, 6, 10, 32, 60, 60, 32, 10, 19, 55, 129, 140, 129, 55, 19, 28, 94, 218, 294, 294, 218, 94, 28, 44, 140, 363, 506, 608, 506, 363, 140, 44, 60, 208, 536, 832, 1038, 1038, 832, 536, 208, 60, 85, 285, 785, 1240, 1695

Offset: 1

Christopher Hunt Gribble, Sep 29 2014

			T(n,k) for 1<=n<=9 and 1<=k<=9 is:
   k    1     2     3     4     5     6     7     8     9 ...
n
1       0     0     1     2     6    10    19    28    44
2       0     1     6    14    32    55    94   140   208
3       1     6    27    60   129   218   363   536   785
4       2    14    60   140   294   506   832  1240  1802
5       6    32   129   294   608  1038  1695  2516  3642
6      10    55   218   506  1038  1785  2902  4324  6242
7      19    94   363   832  1695  2902  4703  6992 10075
8      28   140   536  1240  2516  4324  6992 10416 14988
9      44   208   785  1802  3642  6242 10075 14988 21544

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

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248016, A248059, A248060, A248017, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)*(1/96);
end proc;
f := seq(seq(b(n, k - n + 1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)/96;
T(1,k) = A005993(k-3) = (k-1)*(2*(k-2)*k + 3*(1-(-1)^k))/24;
T(2,k) = A225972(k) = (k-1)*(2*k*(2*k-1) + 3*(1-(-1)^k))/12;
T(2,k) - T(1,k) = A199771(k-1) and A212561(k) = (k-1)*(6*k^2 + 3*(1-(-1)^k))/24.

Terms corrected and extended by Christopher Hunt Gribble, Apr 01 2015

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

A248059 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing four 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, 1, 0, 1, 6, 6, 1, 3, 22, 39, 22, 3, 9, 60, 139, 139, 60, 9, 19, 135, 371, 476, 371, 135, 19, 38, 266, 813, 1253, 1253, 813, 266, 38, 66, 476, 1574, 2706, 3254, 2706, 1574, 476, 66, 110, 792, 2770, 5199, 6969, 6969, 5199, 2770, 792, 110, 170, 1245

Offset: 1

Christopher Hunt Gribble, Sep 30 2014

			T(n,k) for 1<=n<=9 and 1<=k<=9 is:
   k    1      2      3      4      5      6      7      8       9 ...
n
1       0      0      0      1      3      9     19     38      66
2       0      1      6     22     60    135    266    476     792
3       0      6     39    139    371    813   1574   2770    4554
4       1     22    139    476   1253   2706   5199   9080   14857
5       3     60    371   1253   3254   6969  13294  23102   37637
6       9    135    813   2706   6969  14841  28197  48852   79401
7      19    266   1574   5199  13294  28197  53381  92266  149645
8      38    476   2770   9080  23102  48852  92266 159216  257878
9      66    792   4554  14857  37637  79401 149645 257878  417156

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, A248060, A248017, A248027.

b := proc (n::integer, k::integer)::integer;
(4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

Empirically,
T(n,k) = (4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384;
T(1,k) = sum(A005993(i-4),i=1,k)
       = sum((i-2)*(2*(i-3)*(i-1) + 3*(1-(-1)^(i-1)))/24, i=1,k);
T(2,k) = A071239(k-1) = (k-1)*k*((k-1)^2+2)/6.

Terms corrected and extended by Christopher Hunt Gribble, Apr 06 2015

A325181 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 08 2019

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

			The a(2) = 2 through a(15) = 1 partitions:
(2)  (21) (32)  (33)  (322) (332) (433)  (443)  (444)  (4333) (4433) (4443)
(11)      (221) (222) (331)       (3331) (3332) (3333) (4432) (4442)
                (321)                    (4331) (4332) (4441)
                                                (4431)

Giovanni Resta, Table of n, a(n) for n = 0..150

Cf. A006918, A084835, A096771, A257990, A263297, A325178, A325179, A325182, A325191, A325192, A325198.

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==1&]],{n,0,30}]

More terms from Giovanni Resta, Apr 15 2019

cp's OEIS Frontend

A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A325180 Heinz number of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A059594 Convolution triangle based on A008619 (positive integers repeated).

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Maxima

Formula

A243866 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing one 1 X 1 tile in an n X k rectangle under all symmetry operations of the rectangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A244306 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing two 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A248011 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing three 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).