A008763 -id:A008763 - cp's OEIS frontend

A224697 Number A(n,k) of different ways to divide an n X k rectangle into subsquares, considering only the list of parts; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 4, 5, 7, 5, 4, 1, 1, 1, 1, 4, 7, 9, 9, 7, 4, 1, 1, 1, 1, 5, 8, 14, 11, 14, 8, 5, 1, 1, 1, 1, 5, 10, 17, 20, 20, 17, 10, 5, 1, 1

Offset: 0

Alois P. Heinz, Apr 15 2013

			A(4,5) = 9 because there are 9 ways to divide a 4 X 5 rectangle into subsquares, considering only the list of parts: [20(1 X 1)], [16(1 X 1), 1(2 X 2)], [12(1 X 1), 2(2 X 2)], [11(1 X 1), 1(3 X 3)], [8(1 X 1), 3(2 X 2)], [7(1 X 1), 1(2 X 2), 1(3 X 3)], [4(1 X 1), 4(2 X 2)], [4(1 X 1), 1(4 X 4)], [3(1 X 1), 2(2 X 2), 1(3 X 3)].  There is no way to divide this rectangle into [2(1 X 1), 2(3 X 3)].
Square array A(n,k) begins:
  1, 1, 1,  1,  1,  1,  1,   1,   1,   1, ...
  1, 1, 1,  1,  1,  1,  1,   1,   1,   1, ...
  1, 1, 2,  2,  3,  3,  4,   4,   5,   5, ...
  1, 1, 2,  3,  4,  5,  7,   8,  10,  12, ...
  1, 1, 3,  4,  7,  9, 14,  17,  24,  29, ...
  1, 1, 3,  5,  9, 11, 20,  26,  36,  48, ...
  1, 1, 4,  7, 14, 20, 31,  47,  71,  95, ...
  1, 1, 4,  8, 17, 26, 47,  57, 102, 143, ...
  1, 1, 5, 10, 24, 36, 71, 102, 148, 238, ...
  1, 1, 5, 12, 29, 48, 95, 143, 238, 312, ...

Alois P. Heinz and Christopher Hunt Gribble, Antidiagonals n = 0..27, flattened (first 25 antidiagonals from Alois P. Heinz)

Columns (or rows) k=0+1, 2-5 give: A000012, A008619, A001399, A008763(n+4), A187753.
Main diagonal gives: A034295.
Cf. A225622.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then {} elif n=0 or l=[] then {[]}
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:={};
         for i from k to nops(l) while l[i]=0 do s:=s union
             map(x->sort([x[], 1+i-k]), b(n, [l[j]$j=1..k-1,
                 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, nops(b(n, [0$k])), nops(b(k, [0$n]))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, l_] := b[n, l] = Module[{i, k, m, s, t}, Which[Max[l] > n, {}, n == 0 || l == {}, {{}}, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[x, Sort[Append[x, 1+i-k]]], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]] ] ]]]; s]]; a[n_, k_] := If[n >= k, Length @ b[n, Array[0&, k]], Length @ b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

A089299 Number of square plane partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 41, 57, 78, 108, 146, 202, 274, 375, 509, 690, 929, 1255, 1679, 2246, 2991, 3979, 5266, 6971, 9187, 12104, 15898, 20870, 27322, 35762, 46690, 60927, 79348, 103270, 134138, 174108, 225576, 291990, 377320, 487083

Offset: 0

N. J. A. Sloane, Dec 25 2003

nonn

Number of ways of writing n as a sum p(1,1) + p(1,2) + ... + p(1,k) + p(2,1) + ... + p(2,k) + ... + p(k,1) + ... + p(k,k) for some k so that in the square array {p(i,j)} the numbers are nonincreasing along rows and columns. All the p(i,j) are >= 1.

			a(7) = 5:
7 41 32 31 22
. 11 11 21 21
a(10) = 16 from {{10}}, {{3, 2}, {3, 2}}, {{3, 3}, {2, 2}}, {{3, 3}, {3, 1}}, {{4, 1}, {4, 1}}, {{4, 2}, {2, 2}}, {{4, 2}, {3, 1}}, {{4, 3}, {2, 1}}, {{4, 4}, {1, 1}}, {{5, 1}, {3, 1}}, {{5, 2}, {2, 1}}, {{5, 3}, {1, 1}}, {{6, 1}, {2, 1}}, {{6, 2}, {1, 1}}, {{7, 1}, {1, 1}}, {{2, 1, 1}, {1, 1, 1}, {1, 1, 1}}
From _Gus Wiseman_, Jan 16 2019: (Start)
The a(10) = 16 square plane partitions:
  [ten]
.
  [32] [33] [33] [41] [42] [42] [43] [44] [51] [52] [53] [61] [62] [71]
  [32] [22] [31] [41] [22] [31] [21] [11] [31] [21] [11] [21] [11] [11]
.
  [211]
  [111]
  [111]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A008763, A001970, A089292.

Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 16 2019 *)

G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..2k-1} (1-x^j)^min(j,2k-j). - Franklin T. Adams-Watters, Jun 14 2006

Corrected and extended by Wouter Meeussen, Dec 30 2003
a(21)-a(25) from John W. Layman, Jan 02 2004
More terms from Franklin T. Adams-Watters, Jun 14 2006
Name edited by Gus Wiseman, Jan 16 2019

A266769 Expansion of 1/((1-x)(1-x^2)^2(1-x^3)).

Original entry on oeis.org

1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061

Offset: 0

N. J. A. Sloane, Jan 10 2016

This is the same as A008763 but without the four leading zeros. There are so many situations where one wants this sequence rather than A008763 that it seems appropriate for it to have its own entry.
But see A008763 (still the main entry) for numerous applications and references.
Also, Molien series for invariants of finite Coxeter group D_4 (bisected).
The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017
a(n) is the number of partitions of n into parts 1, 2, and 3, where there are two sorts of parts 2. - Joerg Arndt, Jun 27 2017

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.

A variant of A008763.

I:=[1,1,3,4,7,9,14,17]; [n le 8 select I[n] else Self(n-1)+2*Self(n-2)-Self(n-3)-2*Self(n-4)-Self(n-5)+2*Self(n-6)+Self(n-7)-Self(n-8): n in [1..60]]; // Vincenzo Librandi, Jan 11 2016

CoefficientList[Series[1/((1-x)*(1-x^2)^2*(1-x^3)), {x, 0, 50}], x] (* JungHwan Min, Jan 10 2016 *)
LinearRecurrence[{1, 2, -1, -2, -1, 2, 1, -1}, {1, 1, 3, 4, 7, 9, 14, 17}, 100] (* Vincenzo Librandi, Jan 11 2016 *)

Vec(1/((1-x)*(1-x^2)^2*(1-x^3)) + O(x^100)) \\ Michel Marcus, Jan 11 2016

{a(n) = (9*(n+4)*(-1)^n + 2*n^3 + 24*n^2 + 87*n + 157) \ 144}; /* Michael Somos, Jun 26 2017 */

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) for n>7. - Vincenzo Librandi, Jan 11 2016

a(n) = -a(-8-n) for all n in Z. - Michael Somos, Jun 26 2017

A097701 Expansion of 1/((1-x)^2(1-x^2)^2(1-x^3)).

Original entry on oeis.org

1, 2, 5, 9, 16, 25, 39, 56, 80, 109, 147, 192, 249, 315, 396, 489, 600, 726, 874, 1040, 1232, 1446, 1690, 1960, 2265, 2600, 2975, 3385, 3840, 4335, 4881, 5472, 6120, 6819, 7581, 8400, 9289, 10241, 11270, 12369, 13552, 14812, 16164, 17600, 19136

Offset: 0

Ralf Stephan, Aug 24 2004

Number of partitions of 5*n+12 or 5*n+13 into 5 parts (+-) 3 mod 5. For example, the a(3) = 9 partitions of 27 are: [18,3,2,2,2], [13,8,2,2,2], [17,3,3,2,2], [12,7,3,3,2], [7,7,7,3,3], [13,7,3,2,2], [8,8,7,2,2], [12,8,3,2,2], [8,7,7,3,2]. - Richard Turk, Apr 23 2016

Number of partitions of n into two kinds of parts 1, two kinds of parts 2, and one kind of parts 3. - Joerg Arndt, Apr 24 2016

			G.f. = 1 + 2*x + 5*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 39*x^6 + ... - _Michael Somos_, Aug 16 2023

Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

First differences of A002625. Partial sums of A008763.

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=1)}, unlabeled]: subs(r=5,stack): seq(count(subs(r=3,ZL),size=m),m=3..47) ; # Zerinvary Lajos, Mar 09 2007

CoefficientList[Series[1/((1-x)^2(1-x^2)^2(1-x^3)),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{1,2,5,9,16,25,39,56,80},50] (* Harvey P. Dale, May 20 2013 *)
a[ n_] := Round[(n + 1)*(9*(-1)^n + n^3 + 17*n^2 + 95*n + 184)/288]; (* Michael Somos, Aug 16 2023*)

a(n)=1/576*(2*n^4+36*n^3+224*n^2+558*n+495+(18*n+81)*(-1)^n-64*(if(n%3,1,0)))

x='x+O('x^99); Vec(1/((1-x)^2*(1-x^2)^2*(1-x^3))) \\ Altug Alkan, Sep 18 2016

a(n) = floor((n + 1) * (9*(-1)^n + n^3 + 17*n^2 + 95*n + 184)/288 + 1/2). - Tani Akinari, Oct 07 2012
a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) - a(n-4) + a(n-5) + 3*a(n-6) - a(n-7) - 2*a(n-8) + a(n-9) for n >= 9, with initial values as shown. - Harvey P. Dale, May 20 2013
a(n) = (6*n*(9*((-1)^n + 31) + n*(n*(n + 18) + 112)) + 243*(-1)^n + 128*cos((2*Pi*n)/3) + 1357)/1728. - Ilya Gutkovskiy, Apr 23 2016
a(n) = 1 + 175*n/288 + 47*n^2/144 + n^3/16 + n^4/288 + (9/16 + n/8)*floor(n/2) + 2*floor(n/3)/9 + floor((n+1)/3)/9. - Vaclav Kotesovec, Apr 24 2016
a(n) = a(-9-n) for all n in Z. - Michael Somos, Aug 16 2023

A008731 Molien series for 3-dimensional group [2, n] = *22n.

Original entry on oeis.org

1, 0, 2, 1, 3, 2, 5, 3, 7, 5, 9, 7, 12, 9, 15, 12, 18, 15, 22, 18, 26, 22, 30, 26, 35, 30, 40, 35, 45, 40, 51, 45, 57, 51, 63, 57, 70, 63, 77, 70, 84, 77, 92, 84, 100, 92, 108, 100, 117, 108, 126, 117, 135, 126, 145, 135, 155, 145, 165, 155, 176, 165, 187

Offset: 0

N. J. A. Sloane

nonn

a(n+4) is the number of solutions to the equation X + Y + Z = n such that X < Z, Y < Z, and X + Y >= Z. - Geoffrey Critzer, Jul 13 2013

Number of partitions of n into two sorts of 2, and one sort of 3. - Joerg Arndt, Jul 14 2013

			a(4) = 3 because we have:
1 + 3 + 4 = 2 + 2 + 4 = 3 + 1 + 4. - _Geoffrey Critzer_, Jul 13 2013
G.f. = 1 + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 5*x^6 + 3*x^7 + 7*x^8 + 5*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 222
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).

First differences of A008763.

a:=[1,0,2,1,3,2,5];; for n in [8..70] do a[n]:=2*a[n-2]+a[n-3]-a[n-4]-2*a[n-5]+a[n-7]; od; a; # G. C. Greubel, Jul 30 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Jul 30 2019

seq(coeff(series(1/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0 .. 70); # modified by G. C. Greubel, Jul 30 2019

CoefficientList[Series[1/(1-x^2)^2/(1-x^3),{x,0,70}],x] (* Geoffrey Critzer, Jul 13 2013 *)
a[ n_] := Quotient[ (2 n^2 + If[ OddQ[n], 8 n + 6, 20 n + 48]), 70]; (* Michael Somos, Feb 02 2015 *)
a[ n_] := Module[{m=n}, If[ n < 0, m=-7-n]; SeriesCoefficient[ 1 / ( (1 - x^2)^2 * (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Feb 02 2015 *)
LinearRecurrence[{0,2,1,-1,-2,0,1},{1,0,2,1,3,2,5},70] (* Harvey P. Dale, Feb 23 2018 *)

{a(n) = (2*n^2 + if( n%2, 8*n + 6, 20*n + 48)) \ 48}; /* Michael Somos, Feb 02 2015 */

{a(n) = if( n<0, n=-7-n); polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3)) + x * O(x^n), n)}; /* Michael Somos, Feb 02 2015 */

(1/((1-x^2)^2*(1-x^3))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f.: 1/((1-x^2)^2*(1-x^3)) = 1/((1-x)^3*(1+x)^2*(1+x+x^2)).
a(n) = (1/48)*(2*n^2 + 14*n + 27 + (6*n+21)*(-1)^n - 16(n=1 mod 3)).
Euler transform of length 3 sequence [ 0, 2, 1]. - Michael Somos, Feb 02 2015
a(n) = a(-7-n) for all n in Z. - Michael Somos, Feb 02 2015
0 = a(n) + a(n+1) - a(n+2) - 2*a(n+3) - a(n+4) + a(n+5) + a(n+6) - 1 for all n in Z. - Michael Somos, Feb 02 2015
a(n+3) - a(n) = 0 if n even, (n+5)/2 otherwise. - Michael Somos, Feb 02 2015
|a(n)-a(n-1)| = A154958(n). - R. J. Mathar, Aug 11 2021

A060999 Nearest integer to (n+1)^3/9.

Original entry on oeis.org

0, 1, 3, 7, 14, 24, 38, 57, 81, 111, 148, 192, 244, 305, 375, 455, 546, 648, 762, 889, 1029, 1183, 1352, 1536, 1736, 1953, 2187, 2439, 2710, 3000, 3310, 3641, 3993, 4367, 4764, 5184, 5628, 6097, 6591, 7111, 7658, 8232, 8834, 9465, 10125, 10815, 11536

Offset: 0

N. J. A. Sloane, May 14 2001

a(n) is also the number of ways to award 4n+5 bonuses to 4 teams: first, second, third and fourth satisfying 1st > 2nd > 3rd > 4th and 1st + 4th < 2nd + 3rd. - Hoang Xuan Thanh, Jun 03 2025

			x + 3*x^2 + 7*x^3 + 14*x^4 + 24*x^5 + 38*x^6 + 57*x^7 + 81*x^8 + ...

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

Cf. A266769, A008763.

Table[Floor[(n+1)^3/9+1/2],{n,0,50}] (* Harvey P. Dale, Jan 20 2013 *) (* or *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 1, 3, 7, 14, 24}, 47] (* Georg Fischer, Oct 13 2020 *)

a(n) = { round((n + 1)^3/9) } \\ Harry J. Smith, Jul 16 2009

{a(n) = n++; (n^3 - kronecker(-3, n)) / 9} /* Michael Somos, Aug 12 2009 */

G.f.: x*(1+x^2)/((1-x)^3*(1-x^3)).
G.f.: x * (1 - x^4) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)).
G.f.: ( (1 + 4*x + x^2) / (1 - x)^4 - 1 / (1 + x + x^2) ) / 9.
From Michael Somos, Aug 12 2009: (Start)
Euler transform of length 4 sequence [ 3, 1, 1, -1].
a(-2-n) = -a(n). (End)
E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(1 + 7*x + 6*x^2 + x^3) - 3*cos(sqrt(3)*x/2)+ sqrt(3)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Sep 24 2024
From Hoang Xuan Thanh, Jun 03 2025: (Start)
a(n) = floor(((n+1)^3+1)/9).
For n>0: a(n) = A266769(2n-2). (End)

A069950 Expansion of (1+x^2)(1+x^5)(1+x^8)/((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)(1-x^6)(1-x^7)(1-x^8)(1-x^9)*(1-x^10)).

Original entry on oeis.org

1, 1, 3, 4, 7, 11, 17, 25, 38, 53, 77, 105, 146, 196, 265, 350, 462, 600, 778, 994, 1270, 1601, 2016, 2514, 3126, 3857, 4745, 5797, 7063, 8554, 10331, 12411, 14871, 17734, 21093, 24986, 29519, 34747, 40801, 47746, 55746, 64884, 75353

Offset: 0

N. J. A. Sloane, May 05 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242.
Index entries for linear recurrences with constant coefficients, signature (1, 2, -1, -2, 1, 1, -3, -1, 3, 1, -3, 2, 4, -1, -2, 1, -1, -3, -1, 0, 0, 0, 2, 2, 2, 0, 0, 0, -1, -3, -1, 1, -2, -1, 4, 2, -3, 1, 3, -1, -3, 1, 1, -2, -1, 2, 1, -1).

Cf. A008763, A060962.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^2)*(1+x^5)*(1+x^8)/( (&*[1-x^j: j in [1..10]]) ) )); // G. C. Greubel, Aug 16 2022

CoefficientList[Series[(1+x^2)*(1+x^5)*(1+x^8)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/ (1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)/(1-x^9)/(1-x^10),{x,0,60}],x] (* Harvey P. Dale, Feb 25 2013 *)

Vec((1+x^2)*(1+x^5)*(1+x^8)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)/(1-x^9)/(1-x^10)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

def A069950_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)*(1+x^5)*(1+x^8)/product(1-x^j for j in (1..10)) ).list()
A069950_list(60) # G. C. Greubel, Aug 16 2022

G.f.: (1+x^2)*(1+x^5)*(1+x^8)/( Product_{j=1..10} (1-x^j) ). - G. C. Greubel, Aug 16 2022

A070557 Number of two-rowed partitions of length 4.

Original entry on oeis.org

1, 1, 3, 5, 10, 15, 26, 38, 60, 85, 125, 172, 243, 325, 442, 580, 767, 986, 1275, 1612, 2045, 2548, 3179, 3910, 4812, 5849, 7109, 8554, 10285, 12259, 14599, 17255, 20372, 23895, 27991, 32603, 37925, 43890, 50725, 58361, 67053, 76727, 87678, 99825, 113503

Offset: 0

N. J. A. Sloane, May 07 2002

nonn

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.

Cf. A008763, A001993, A070558, A070559.

a:= n-> (Matrix(24, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 2, 0, -1, -4, -2, 1, 5, 6, 0, -4, -6, -4, 0, 6, 5, 1, -2, -4, -1, 0, 2, 1, -1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..50); # Alois P. Heinz, Jul 31 2008

m = 4; n = 45; gf = 1/((1-x)*Product[1-x^k, {k, 2, m}]^2*(1-x^(m+1))) + O[x]^n; CoefficientList[gf, x] (* Jean-François Alcover, Jul 17 2015 *)

G.f.: 1/((1-x)*((1-x^2)*...*(1-x^m))^2*(1-x^(m+1))) for m = 4.

A070558 Number of two-rowed partitions of length 5.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 28, 42, 68, 100, 151, 215, 312, 432, 605, 821, 1117, 1485, 1977, 2581, 3371, 4335, 5566, 7060, 8938, 11196, 13994, 17338, 21426, 26280, 32152, 39074, 47369, 57093, 68637, 82097, 97955, 116339, 137849, 162665, 191507

Offset: 0

N. J. A. Sloane, May 07 2002

nonn

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.

Cf. A008763, A001993, A070557, A070559.

a:= n-> (Matrix(35, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 2, 0, -1, -3, -2, -2, 3, 7, 5, 1, -4, -8, -11, -1, 5, 9, 9, 5, -1, -11, -8, -4, 1, 5, 7, 3, -2, -2, -3, -1, 0, 2, 1, -1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..40); # Alois P. Heinz, Jul 31 2008

m = 5; n = 45; gf = 1/((1-x)*Product[1-x^k, {k, 2, m}]^2*(1-x^(m+1))) + O[x]^n; CoefficientList[gf, x] (* Jean-François Alcover, Jul 17 2015 *)

G.f.: 1/((1-x)*((1-x^2)*...*(1-x^m))^2*(1-x^(m+1))) for m = 5.

A070559 Number of two-rowed partitions of length 6.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 29, 44, 72, 108, 166, 241, 357, 504, 720, 998, 1386, 1882, 2559, 3413, 4551, 5981, 7842, 10162, 13138, 16811, 21454, 27150, 34251, 42898, 53570, 66464, 82221, 101146, 124057, 151404, 184261, 223235, 269723, 324578

Offset: 0

N. J. A. Sloane, May 07 2002

nonn

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.

Cf. A008763, A001993, A070557, A070558.

a:= n-> (Matrix(48, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 2, 0, -1, -3, -1, -2, 0, 5, 6, 5, 1, -5, -11, -9, -7, 2, 9, 15, 16, 4, -5, -13, -16, -13, -5, 4, 16, 15, 9, 2, -7, -9, -11, -5, 1, 5, 6, 5, 0, -2, -1, -3, -1, 0, 2, 1, -1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..39); # Alois P. Heinz, Jul 31 2008

m = 6; n = 40; gf = 1/((1-x)*Product[1-x^k, {k, 2, m}]^2*(1-x^(m+1))) + O[x]^n; CoefficientList[gf, x] (* Jean-François Alcover, Jul 17 2015 *)

G.f.: 1/((1-x)*((1-x^2)*...*(1-x^m))^2*(1-x^(m+1))) for m = 6.

cp's OEIS Frontend

A224697 Number A(n,k) of different ways to divide an n X k rectangle into subsquares, considering only the list of parts; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

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A089299 Number of square plane partitions of n.

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A266769 Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)).

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A097701 Expansion of 1/((1-x)^2*(1-x^2)^2*(1-x^3)).

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A008731 Molien series for 3-dimensional group [2, n] = *22n.

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A060999 Nearest integer to (n+1)^3/9.

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A266769 Expansion of 1/((1-x)(1-x^2)^2(1-x^3)).

A097701 Expansion of 1/((1-x)^2(1-x^2)^2(1-x^3)).

A069950 Expansion of (1+x^2)(1+x^5)(1+x^8)/((1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)(1-x^6)(1-x^7)(1-x^8)(1-x^9)*(1-x^10)).