A026810 -id:A026810 - cp's OEIS frontend

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

0, 0, 0, 0, 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, 2187, 2304, 2439

Offset: 0

N. J. A. Sloane, Simon Plouffe, Stephen P. Humphries

Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(n+4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki, Dec 22 2005
Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - Michael Somos, Jan 21 2015
Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - Wesley Ivan Hurt, Jan 19 2021

			a(7) = 4:
41 32 31 22
11 11 21 21
G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...
a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015
a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015
a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=1).
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Michael Somos, In the Elliptic Realm
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

See A266769 for a version without the four leading zeros.
Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.

a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do 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]; od; a; # G. C. Greubel, Sep 10 2019

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(H);

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 10 2019

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,2,-1,-2,-1,2,1,-1][i] else 0 fi)^n)[1,5]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* Jean-François Alcover, Mar 30 2011 *)
LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* Harvey P. Dale, Mar 04 2012 *)
a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* Michael Somos, Jan 21 2015 *)
a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* Michael Somos, Jan 21 2015 *)

{a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* Michael Somos, Jan 21 2015 */

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

def AA008763_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()
AA008763_list(60) # G. C. Greubel, Sep 10 2019

Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).
a(n) = (1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*Pi*n/3). - Richard Choulet, Nov 27 2008
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), n>7. - Harvey P. Dale, Mar 04 2012
a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012
a(n+1) - a(n) = A008731(n-3). - R. J. Mathar, Aug 06 2013
a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 21 2015
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017

Entry revised Dec 25 2003

A026812 Number of partitions of n in which the greatest part is 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760

Offset: 0

Clark Kimberling

Also number of partitions of n into 6 parts. - Washington Bomfim, Jan 15 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
G. E. Andrews, Partitions: At the Interface of q-Series and Modular Forms, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Essentially same as A001402.
Cf. A026810, A026811, A026813, A026814, A026815, A026816.
Cf. A001401, A001402.

List([0..70],n->NrPartitions(n,6)); # Muniru A Asiru, May 17 2018

Table[ Length[ Select[ Partitions[n], First[ # ] == 6 & ]], {n, 1, 60} ]
CoefficientList[Series[x^6/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Drop[LinearRecurrence[{1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1}, Append[Table[0,{20}],1],115],14] (* Robert A. Russell, May 17 2018 *)

my(x='x+O('x^99)); concat(vector(6), Vec(x^6/prod(k=1, 6, 1-x^k))) \\ Altug Alkan, May 17 2018

a = vector(60,n,n--; round((n+11)*((6*n^4+249*n^3+2071*n^2 -4931*n+40621) /518400 +n\2*(n+10)/192+((n+1)\3+n\3*2)/54))); a = concat([0,0,0,0,0,0], a) \\ Washington Bomfim, Jan 16 2021

G.f.: x^6 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - Colin Barker, Dec 20 2012
a(n) = A008284(n,6). - Robert A. Russell, May 13 2018
a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} 1. - Wesley Ivan Hurt, Jun 29 2019
a(n) = A001402(n) - A001401(n). a(n) = A001402(n-6). - Washington Bomfim, Jan 15 2021
a(n) = round((1/86400)*n^5 + (1/3840)*n^4 + (19/12960)*n^3 - (n mod 2)*(1/384)*n^2 + (1/17280)*b(n mod 6)*n), where b(0)=96, b(1)=b(5)=-629, b(2)=b(4)=-224, and b(3)=-309. - Washington Bomfim and Jon E. Schoenfield, Jan 16 2021

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 prepended by Seiichi Manyama, Jun 08 2017

A062890 Number of quadrilaterals that can be formed with perimeter n. In other words, number of partitions of n into four parts such that the sum of any three is more than the fourth.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 12, 16, 18, 23, 24, 31, 33, 41, 43, 53, 55, 67, 69, 83, 86, 102, 104, 123, 126, 147, 150, 174, 177, 204, 207, 237, 241, 274, 277, 314, 318, 358, 362, 406, 410, 458, 462, 514, 519, 575, 579, 640, 645, 710

Offset: 0

Amarnath Murthy, Jun 29 2001

Partition sets of n into four parts (sides) such that the sum of any three is more than the fourth do not uniquely define a quadrilateral, even if it is further constrained to be cyclic. This is because the order of adjacent sides is important. E.g. the partition set [1,1,2,2] for a perimeter n=6 can be reordered to generate two non-congruent cyclic quadrilaterals, [1,2,1,2] and [1,1,2,2], where the first is a rectangle and the second a kite. - Frank M Jackson, Jun 29 2012

			a(7) = 2 as the two partitions are (1,2,2,2), (1,1,2,3) and in each sum of any three is more than the fourth.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 19.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,0,0,-1,1,-1,1,1,-1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), this sequence (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

CoefficientList[Series[x^4*(1+x+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)), {x, 0, 60}], x] (* Frank M Jackson, Jun 09 2017 *)

G.f.: x^4*(1+x+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).
a(2*n+6) = A026810(2*n+6) - A000601(n), a(2*n+7) = A026810(2*n+7) - A000601(n) for n >= 0. - Seiichi Manyama, Jun 08 2017
From Wesley Ivan Hurt, Jan 01 2021: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-8) + a(n-9) - a(n-10) + a(n-11) + a(n-12) - a(n-13).
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign(floor((i+j+k)/(n-i-j-k+1))). (End)

More terms from Vladeta Jovovic and Dean Hickerson, Jul 01 2001

A024788 Number of 4's in all partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, 105, 139, 190, 250, 336, 436, 575, 740, 963, 1228, 1577, 1995, 2538, 3186, 4013, 5005, 6256, 7751, 9617, 11847, 14605, 17894, 21927, 26730, 32582, 39531, 47942, 57915, 69920, 84114, 101116, 121176, 145095, 173248

Offset: 1

Clark Kimberling

The sums of four successive terms give A000070. - Omar E. Pol, Jul 12 2012
a(n) is also the difference between the sum of 4th largest and the sum of 5th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(n+4) is the number of n-vertex graphs that do not contain a triangle, P_4, or K_2,3 as induced subgraph. These are the K_2,3-free bipartite cographs. Bipartite cographs are graph that are disjoint unions of complete bipartite graphs [Babel et al. Corollary 2.2], and forbidding K_2,3 leaves one possible component for each size except size 4, where there are two. Thus, this number is A000041(n) + a(n) = a(n+4). - Falk Hüffner, Jan 11 2016
a(n) (n>=3) is the number of even singletons in all partitions of n-2 (by a singleton we mean a part that occurs exactly once). Example: a(7) = 3 because in the partitions [5], [4*,1], [3,2*], [3,1,1], [2,2,1], [2*,1,1,1], [1,1,1,1,1] we have 3 even singletons (marked by *). The statement of this comment can be obtained by setting k=2 in Theorem 2 of the Andrews et al. reference. - Emeric Deutsch, Sep 13 2016

			From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 7 we have:
--------------------------------------
.                             Number
Partitions of 7               of 4's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 0
6 + 1 .......................... 0
3 + 3 + 1 ...................... 0
4 + 2 + 1 ...................... 1
2 + 2 + 2 + 1 .................. 0
5 + 1 + 1 ...................... 0
3 + 2 + 1 + 1 .................. 0
4 + 1 + 1 + 1 .................. 1
2 + 2 + 1 + 1 + 1 .............. 0
3 + 1 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
.           7 - 4 =              3
The difference between the sum of the fourth column and the sum of the fifth column of the set of partitions of 7 is 7 - 4 = 3 and equals the number of 4's in all partitions of 7, so a(7) = 3.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews and E. Deutsch, A note on a method of Erdos and the Stanley-Elder theorems, Integers, 16 (2016), A24.
L. Babel, A. Brandstädt, and V. B. Le, Recognizing the P4-structure of bipartite graphs, Discrete Appl. Math. 93 (1999), 157-168.
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Cf. A000041, A066633, A024786, A024787, A024789, A024790, A024791, A024792, A024793, A024794.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0$2], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+`if`(i=4, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Table[ Count[ Flatten[ IntegerPartitions[n]], 4], {n, 1, 50} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 4, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

a(n) = A181187(n,4) - A181187(n,5). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 26 2013: (Start)
a(n+4) - a(n) = A000041(n). a(n) + a(n+2) = A024786(n).
O.g.f.: x^4/(1 - x^4) * product {k >= 1} 1/(1 - x^k) = x^4 + x^5 + 2*x^6 + 3*x^7 + ....
Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)) * (1 - 49*Pi/(24*sqrt(6*n)) + (49/48 + 1633*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016
G.f.: x^4/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)) * Sum_{n >= 0} x^(4*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A026810 (partitions into 4 parts, or, modulo offset differences, partitions into parts <= 4) and A008484 (partitions into parts >= 4). - Peter Bala, Jan 17 2021

A152146 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of 2n into 2k odd parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 1, 0, 3, 3, 2, 1, 1, 0, 3, 5, 3, 2, 1, 1, 0, 4, 6, 5, 3, 2, 1, 1, 0, 4, 9, 7, 5, 3, 2, 1, 1, 0, 5, 11, 11, 7, 5, 3, 2, 1, 1, 0, 5, 15, 14, 11, 7, 5, 3, 2, 1, 1, 0, 6, 18, 20, 15, 11, 7, 5, 3, 2, 1, 1, 0, 6, 23, 26, 22, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 0

R. J. Mathar, Sep 25 2009, indices corrected Jul 09 2012

In both this and A152157, reading columns downwards "converges" to A000041.

Also the number of strict integer partitions of 2n with alternating sum 2k. Also the number of normal integer partitions of 2n of which 2k parts are odd, where a partition is normal if it covers an initial interval of positive integers. - Gus Wiseman, Jun 20 2021

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  1   1
  0  2  2   1   1
  0  3  3   2   1   1
  0  3  5   3   2   1   1
  0  4  6   5   3   2   1  1
  0  4  9   7   5   3   2  1  1
  0  5 11  11   7   5   3  2  1  1
  0  5 15  14  11   7   5  3  2  1  1
  0  6 18  20  15  11   7  5  3  2  1  1
  0  6 23  26  22  15  11  7  5  3  2  1  1
  0  7 27  35  29  22  15 11  7  5  3  2  1  1
  0  7 34  44  40  30  22 15 11  7  5  3  2  1 1
  0  8 39  58  52  42  30 22 15 11  7  5  3  2 1 1
  0  8 47  71  70  55  42 30 22 15 11  7  5  3 2 1 1
  0  9 54  90  89  75  56 42 30 22 15 11  7  5 3 2 1 1
  0  9 64 110 116  97  77 56 42 30 22 15 11  7 5 3 2 1 1
  0 10 72 136 146 128 100 77 56 42 30 22 15 11 7 5 3 2 1 1
From _Gus Wiseman_, Jun 20 2021: (Start)
For example, row n = 6 counts the following partitions (B = 11):
  (75)  (3333)  (333111)  (33111111)  (3111111111)  (111111111111)
  (93)  (5331)  (531111)  (51111111)
  (B1)  (5511)  (711111)
        (7311)
        (9111)
The corresponding strict partitions are:
  (7,5)      (8,4)      (9,3)    (10,2)   (11,1)  (12)
  (6,5,1)    (5,4,3)    (7,3,2)  (9,2,1)
  (5,4,2,1)  (6,4,2)    (8,3,1)
             (7,4,1)
             (6,3,2,1)
The corresponding normal partitions are:
  43221    33321     3321111    321111111   21111111111  111111111111
  322221   332211    32211111   2211111111
  2222211  432111    222111111
           3222111
           22221111
(End)

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A035294 (row sums), A107379, A152140, A152157.
Column k = 1 is A004526.
Column k = 2-8 is A026810 - A026816.
The non-strict version is A239830.
The reverse non-strict version is A344610.
The reverse version is A344649
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives alternating sum of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A006330, A027187, A116406, A120452, A239829, A343941, A344607, A344608, A344609, A344650, A344651, A344739, A344741.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-2)+`if`(i>n, 0, expand(sqrt(x)*b(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(2*n, 2*n-1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jun 21 2021

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&ats[#]==k&]],{n,0,30,2},{k,0,n,2}] (* Gus Wiseman, Jun 20 2021 *)

T(n,k) = A152140(2n,2k).

A343941 Number of strict integer partitions of 2n with reverse-alternating sum 4.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 3, 4, 5, 7, 8, 10, 11, 14, 15, 18, 20, 23, 25, 29, 31, 35, 38, 42, 45, 50, 53, 58, 62, 67, 71, 77, 81, 87, 92, 98, 103, 110, 115, 122, 128, 135, 141, 149, 155, 163, 170, 178, 185, 194, 201, 210, 218, 227, 235, 245, 253, 263, 272, 282, 291, 302

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts, so a(n) is the number of strict odd-length integer partitions of 2n whose conjugate has exactly 4 odd parts (first example). By conjugation, this is also the number partitions of 2n covering an initial interval and containing exactly four odd parts, one of which is the greatest (second example).

			The a(2) = 1 through a(12) = 10 strict partitions (empty column indicated by dot, A..D = 10..13):
  4   .  521   532   543   653   763     873     983     A93     BA3
               631   642   752   862     972     A82     B92     CA2
                     741   851   961     A71     B81     C91     DA1
                                 64321   65421   65432   76432   76542
                                         75321   75431   76531   86541
                                                 76421   86431   87432
                                                 86321   87421   87531
                                                         97321   97431
                                                                 98421
                                                                 A8321
The a(2) = 1 through a(8) = 5 partitions covering an initial interval:
  1111  .  32111   33211    33321     333221     543211      543321
                   322111   332211    3322211    3332221     5432211
                            3222111   32222111   33222211    33322221
                                                 322222111   332222211
                                                             3222222111

The non-reverse non-strict version is A000710.
The non-reverse version is A026810.
The non-strict version is column k = 2 of A344610.
This is column k = 2 of A344649.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A003242, A006330, A027187, A119899, A152146, A239830, A325535, A344604, A344607, A344608, A344650, A344739.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==4&]],{n,0,30,2}]

More terms from Bert Dobbelaere, Jun 12 2021

A060023 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 4.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, -1, -1, -3, -4, -7, -8, -13, -15, -20, -24, -31, -35, -44, -50, -60, -68, -80, -89, -104, -115, -131, -145, -164, -179, -201, -219, -243, -264, -291, -314, -345, -371, -404, -434, -471, -503, -544, -580, -624, -664, -712, -755, -808, -855, -911, -963, -1024, -1079, -1145

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+3 into 3 parts and the number of partitions of n+3 into 4 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -2, 0, 0, 1, 1, -1).

Cf. A026810, A069905.

Cf. For other values of N: A060022 (N=3), this sequence (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

I:=[1,0,1,1,1,0,1,-1,-1,-3]; [n le 10 select I[n] else Self(n-1)+Self(n-2)-2*Self(n-5)+Self(n-8)+Self(n-9)-Self(n-10): n in [1..60]]; // Vincenzo Librandi, Jun 23 2015

CoefficientList[Series[(1-x-x^4)/Times@@(1-x^Range[4]),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-2,0,0,1,1,-1},{1,0,1,1,1,0,1,-1,-1,-3},70] (* Harvey P. Dale, Jan 14 2015 *)

Vec((1 - x - x^4) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(n) = A069905(n+3) - A026810(n+3). - Wesley Ivan Hurt, Apr 16 2019
From Colin Barker, Apr 17 2019: (Start)
G.f.: (1 - x - x^4) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n>9.
(End)

A060024 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 5.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 0, -3, -3, -8, -10, -16, -20, -29, -35, -47, -56, -72, -85, -105, -122, -148, -171, -202, -231, -270, -306, -353, -397, -453, -507, -573, -637, -715, -791, -881, -970, -1075, -1178, -1298, -1417, -1554, -1691, -1846, -2001, -2177, -2353, -2550, -2748, -2969

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+4 into 4 parts and the number of partitions of n+4 into 5 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

Cf. A026810, A026811.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), this sequence (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

CoefficientList[Series[(1-x-x^5)/(Times@@(1-x^Range[5])),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1},{1,0,1,1,2,1,2,1,2,0,0,-3,-3,-8,-10},60] (* Harvey P. Dale, Dec 21 2015 *)

Vec((1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(0)=1, a(1)=0, a(2)=1, a(3)=1, a(4)=2, a(5)=1, a(6)=2, a(7)=1, a(8)=2, a(9)=0, a(10)=0, a(11)=-3, a(12)=-3, a(13)=-8, a(14)=-10, a(n) = a(n-1)+ a(n-2)-a(n-5)-a(n-6)-a(n-7)+a(n-8)+a(n-9)+a(n-10)-a(n-13)- a(n-14)+ a(n-15). - Harvey P. Dale, Dec 21 2015
a(n) = A026810(n+4) - A026811(n+4). - Wesley Ivan Hurt, Apr 16 2019
G.f.: (1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Apr 17 2019

A211859 Number of partitions of n into parts <= 4 with the property that all parts have distinct multiplicities.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 8, 10, 10, 14, 18, 18, 26, 31, 30, 39, 48, 48, 61, 63, 73, 84, 101, 98, 124, 132, 147, 156, 188, 182, 223, 227, 257, 272, 322, 306, 367, 377, 417, 427, 499, 488, 564, 567, 645, 647, 740, 720, 828, 836, 920, 924, 1048, 1030, 1173, 1161

Offset: 0

Matthew C. Russell, Apr 25 2012

			For n=3 the a(3) = 2 partitions are [3] and [1,1,1]. Note that [2,1] does not count, as 1 and 2 appear with the same nonzero multiplicity.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.

Cf. A026810, A098859.

Cf. A105637, A211858, A211860, A211861, A211862, A211863.

a211859 n = p 0 [] [1..4] n where
   p m ms _      0 = if m `elem` ms then 0 else 1
   p    []     _ = 0
   p m ms ks'@(k:ks) x
     | x < k       = 0
     | m == 0      = p 1 ms ks' (x - k) + p 0 ms ks x
     | m `elem` ms = p (m + 1) ms ks' (x - k)
     | otherwise   = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x
-- Reinhard Zumkeller, Dec 27 2012

G.f.: (9*x^45 +20*x^44 +44*x^43 +76*x^42 +121*x^41 +172*x^40 +234*x^39 +292*x^38 +346*x^37 +380*x^36 +412*x^35 +415*x^34 +417*x^33 +401*x^32 +389*x^31 +365*x^30 +361*x^29 +351*x^28 +359*x^27 +365*x^26 +383*x^25 +391*x^24 +413*x^23 +422*x^22 +436*x^21 +444*x^20 +454*x^19 +454*x^18 +458*x^17 +450*x^16 +437*x^15 +415*x^14 +383*x^13 +342*x^12 +298*x^11 +248*x^10 +198*x^9 +152*x^8 +110*x^7 +76*x^6 +49*x^5 +30*x^4 +16*x^3 +8*x^2 +3*x +1) / ((x^2-x+1) *(x^4-x^3+x^2-x+1) *(x^6+x^3+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+x+1)^2 *(x^4+x^3+x^2+x+1)^2 *(x^2+1)^2 *(x+1)^3 *(x-1)^4). - Alois P. Heinz, Feb 09 2017

cp's OEIS Frontend

A008763 Expansion of g.f.: x^4/((1-x)*(1-x^2)^2*(1-x^3)).

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A026812 Number of partitions of n in which the greatest part is 6.

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A062890 Number of quadrilaterals that can be formed with perimeter n. In other words, number of partitions of n into four parts such that the sum of any three is more than the fourth.

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A024788 Number of 4's in all partitions of n.

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A152146 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of 2n into 2k odd parts.

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A343941 Number of strict integer partitions of 2n with reverse-alternating sum 4.

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A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).