A006918 -id:A006918 - cp's OEIS frontend

A006584 If n mod 2 = 0 then n(n^2-4)/12 else n(n^2-1)/12.

0, 0, 0, 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, 182, 224, 280, 336, 408, 480, 570, 660, 770, 880, 1012, 1144, 1300, 1456, 1638, 1820, 2030, 2240, 2480, 2720, 2992, 3264, 3570, 3876, 4218, 4560, 4940, 5320

Offset: 0

N. J. A. Sloane

Graded dimension of L''/[L',L''] for the free Lie algebra on 2 generators. Let L be a free Lie algebra with 2 generators graded by the total degree. Set L'=[L,L] and L''=[L',L']. Then a(n) is equal to the dimension of the homogeneous subspace of degree n+2 in the quotient L''/[L',L'']. - Sergei Duzhin, Mar 15 2004
Also the 2nd Witt transform of A000027. - R. J. Mathar, Nov 08 2008
Also the number of 3-element subsets of {1..n+1} whose elements sum up to an odd integer, i.e., the third column of A159916: e.g. a(3)=2 corresponds to the two subsets {1,2,4} and {2,3,4} of {1..4}. - M. F. Hasler, May 01 2009
The set of magic numbers for an idealized harmonic oscillator nucleus with a biaxially deformed prolate ellipsoid shape and an oscillator ratio of 2:1. - Jess Tauber, May 13 2013
Quasipolynomial of order 2. - Charles R Greathouse IV, May 14 2013

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.

Moussa Benoumhani and Messaoud Kolli, Finite topologies and partitions, JIS 13 (2010), Article 10.3.5, Lemma 6 6th line.
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000027, A003451, A006918, A027656, A034877.

Partial sums of A110660.

A006584:=n->`if`(n mod 2 = 0, n*(n^2-4)/12, n*(n^2-1)/12): seq(A006584(n), n=0..100); # Wesley Ivan Hurt, May 08 2016

If[EvenQ@ #, #*(#^2 - 4)/12, #*(#^2 - 1)/12] & /@ Range[0, 40] (* or *) Table[n*(2*n^2 - 5 - 3*(-1)^n)/24, {n, 0, 40}] (* Michael De Vlieger, Apr 03 2015 *)

A006584(n)=n*(n^2-if(n%2,1,4))\12 \\ M. F. Hasler, May 01 2009

a(n)=n*if(n%2, n^2-1, n^2-4)/12 \\ Charles R Greathouse IV, Aug 11 2017

a(n+3) = A003451(n) + A027656(n). - Yosu Yurramendi, Aug 07 2008
G.f.: 2*x^3/((1-x)^4*(1+x)^2). a(n) = 2*A006918(n-2). - R. J. Mathar, Nov 08 2008
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Jaume Oliver Lafont, Dec 05 2008
a(n) = n*(2*n^2-5-3*(-1)^n)/24. - Luce ETIENNE, Apr 03 2015
a(n) = Sum_{i=1..n} floor(i*(n-i)/2). - Wesley Ivan Hurt, May 07 2016
E.g.f.: x*(x*(x + 3)*exp(x) - 3*sinh(x))/12. - Ilya Gutkovskiy, May 08 2016
Sum_{n>=3} 1/a(n) = 75/8 - 12*log(2). - Amiram Eldar, Sep 17 2022

A111384 a(n) = binomial(n,3) - binomial(floor(n/2),3) - binomial(ceiling(n/2),3).

Original entry on oeis.org

0, 0, 0, 1, 4, 9, 18, 30, 48, 70, 100, 135, 180, 231, 294, 364, 448, 540, 648, 765, 900, 1045, 1210, 1386, 1584, 1794, 2028, 2275, 2548, 2835, 3150, 3480, 3840, 4216, 4624, 5049, 5508, 5985, 6498, 7030, 7600, 8190, 8820, 9471, 10164, 10879, 11638, 12420, 13248

Offset: 0

N. J. A. Sloane, Nov 10 2005

a(n) is the maximum number of open triangles in a simple, undirected graph with n vertices. - Eugene Lykhovyd, Oct 20 2018
a(n) is the maximum number of elements of the set T := {3} u (IN \ 3IN) that can be written as a sum of three distinct elements of an n-element subset of T, see arXiv link 2309.14840. - Markus Sigg, Sep 27 2023
a(n) is the maximum number of triples (i.e., 3-element subsets of {1..n}) such that there exists a 2-coloring of {1..n} in which no triple is monochromatic.  For the contrasting minimum number of triples such that every 2-coloring of {1..n} results in at least one monochromatic triple, see A385403. - David Dewan, Jul 04 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Peter Keevash and Benny Sudakov, The Turán number of the Fano plane, Combinatorica, 25 (2005), 561-574; alternative link.
Artem Pyatkin, Eugene Lykhovyd, and Sergiy Butenko, The maximum number of induced open triangles in graphs of a given order, Optimization Letters 13, 1927-1935 (2019).
Adityanarayanan Radhakrishnan, Liam Solus, and Caroline Uhler, Counting Markov Equivalence Classes by Number of Immoralities, arXiv preprint arXiv:1611.07493 [math.CO], 2016-2017.
Markus Sigg, A note on OEIS sequence A111384, arXiv:2309.14840 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A006578, A006918.

a:=[0,0,0,1,4,9];; for n in [7..50] do a[n]:=2*a[n-1]+a[n-2]-4*a[n-3]+a[n-4]+2*a[n-5]-a[n-6]; od; a; # Muniru A Asiru, Oct 21 2018

[Binomial(n, 3) - Binomial(Floor(n/2), 3) - Binomial(Ceiling(n/2), 3): n in [0..50]]; // Vincenzo Librandi, Oct 20 2018

seq(floor(n/2)*ceil(n/2)*(n-2)/2,n=0..50); # James R. Buddenhagen, Nov 11 2009

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 0, 0, 1, 4, 9}, 50] (* Vincenzo Librandi, Oct 20 2018 *)

a(n)=floor(n/2)*ceil(n/2)*(n-2)/2 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = floor(n/2)*ceiling(n/2)*(n-2)/2. - James R. Buddenhagen, Nov 11 2009
From R. J. Mathar, Mar 18 2010: (Start)
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: x^3*(1+2*x)/ ((1+x)^2 * (x-1)^4). (End)
a(n) = A006918(n-2) + 2*A006918(n-3). - R. J. Mathar, Jan 20 2018
a(n) = (n-2)*n^2/8 for even n, a(n) = (n-2)*(n^2-1)/8 for odd n. - Markus Sigg, Sep 26 2023
Sum_{n>=3} 1/a(n) = 4/3 - Pi^2/6 + 8*log(2)/3. - Amiram Eldar, Oct 10 2023
E.g.f.: (x + 2)*(x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x))/8. - Stefano Spezia, Apr 08 2024

A144677 Related to enumeration of quantum states (see reference for precise definition).

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 18, 24, 30, 40, 50, 60, 75, 90, 105, 126, 147, 168, 196, 224, 252, 288, 324, 360, 405, 450, 495, 550, 605, 660, 726, 792, 858, 936, 1014, 1092, 1183, 1274, 1365, 1470, 1575, 1680, 1800, 1920, 2040, 2176, 2312, 2448, 2601, 2754, 2907, 3078, 3249, 3420

Offset: 0

N. J. A. Sloane, Feb 06 2009

Equals (1, 2, 3, ...) convolved with (1, 0, 0, 2, 0, 0, 3, ...) = (1 + 2*x + 3*x^2 + ...) * (1 + 2*x^3 + 3*x^6 + ...). - Gary W. Adamson, Feb 23 2010

The Ca2 and Ze4 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of the sequence given above, e.g., Ca2(n) = a(n-1) + 2*a(n-2) + 3*a(n-3) + a(n-4). - Johannes W. Meijer, May 20 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=3]
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A006918, A144678, A144679.

Cf. A000292, A190717. [Johannes W. Meijer, May 20 2011]

I:=[1,2,3,6,9,12,18,24]; [n le 8 select I[n] else 2*Self(n-1)-Self(n-2)+2*Self(n-3)-4*Self(n-4)+2*Self(n-5)-Self(n-6)+2*Self(n-7)-Self(n-8): n in [1..60]]; // Vincenzo Librandi, Mar 28 2015

n:=80; lambda:=3; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b),snc];
od:
S10b;
A144677 := proc(n) option remember; local k1; sum(A190717(n-k1),k1=0..2) end: A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A144677(n), n=0..53); # Johannes W. Meijer, May 20 2011

CoefficientList[Series[1/((x - 1)^4*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Mar 28 2015 *)
LinearRecurrence[{2, -1, 2, -4, 2, -1, 2, -1}, {1, 2, 3, 6, 9, 12, 18, 24}, 60 ] (* Vincenzo Librandi, Mar 28 2015 *)

@CachedFunction
def a(n): return sum( ((j+3)//3)*((n-j+3)//3) for j in (0..n) )
[a(n) for n in (0..60)] # G. C. Greubel, Oct 18 2021

From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A190717(n-2) + A190717(n-1) + A190717(n).
a(n-2) + a(n-1) + a(n) = A014125(n).
G.f.: 1/((1-x)^4*(1+x+x^2)^2). (End)
From Wesley Ivan Hurt, Mar 28 2015: (Start)
a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) +2*a(n-7) -a(n-8).
a(n) = ((2 + floor(n/3))^3 - floor((n+4)/3) + floor((n+4)/3)^3 - floor((n+5)/3) + floor((n+5)/3)^3 - floor((n+6)/3))/6. (End)
a(n) = Sum_{j=0..n} floor((j+3)/3)*floor((n-j+3)/3). - G. C. Greubel, Oct 18 2021
a(n) = (132+129*n+36*n^2+3*n^3+6*(n+5)*cos(2*n*Pi/3)+2*sqrt(3)*(3*n+11)*sin(2*n*Pi/3))/162. - Wesley Ivan Hurt, Sep 01 2025

A325170 Heinz numbers of integer partitions with origin-to-boundary graph-distance equal to 2.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 33, 34, 35, 36, 38, 39, 40, 44, 46, 48, 49, 51, 52, 54, 55, 56, 57, 58, 62, 65, 68, 69, 72, 74, 76, 77, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 104, 106, 108, 111, 112, 115, 116, 118, 119

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps East or South from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the minimum triangular partition contained inside the diagram.

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:
   6: {1,2}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}

Gus Wiseman, Young diagrams corresponding to the first 50 terms.

Cf. A001221, A001222, A006918, A056239, A065770, A112798, A174090, A257990, A297113, A325167, A325169.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[200],otb[Reverse[primeMS[#]]]==2&]

A245558 Square array read by antidiagonals: T(n,k) = number of n-tuples of nonnegative integers (u_0,...,u_{n-1}) satisfying Sum_{j=0..n-1} j*u_j == 1 mod n and Sum_{j=0..n-1} u_j = m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 7, 8, 7, 3, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 20, 25, 20, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset: 1

N. J. A. Sloane, Aug 07 2014

The array is symmetric; for the entries on or below the diagonal see A245559.
If the congruence in the definition is changed from Sum_{j=0..n-1} j*u_j == 1 mod n  to Sum_{j=0..n-1} j*u_j == 0 mod n we get the array shown in A241926, A047996, and A037306.
Differs from A011847 from row n = 9, k = 4 on; if the rows are surrounded by 0's, this yields A051168 without its rows 0 and 1, i.e., a(1) is A051168(2,1). - M. F. Hasler, Sep 29 2018
This array was first studied by Fredman (1975). - Petros Hadjicostas, Jul 10 2019

			Square array begins:
  1, 1,  1,  1,   1,   1,    1,    1,    1,    1, ...
  1, 1,  2,  2,   3,   3,    4,    4,    5,    5, ...
  1, 2,  3,  5,   7,   9,   12,   15,   18,   22, ...
  1, 2,  5,  8,  14,  20,   30,   40,   55,   70, ...
  1, 3,  7, 14,  25,  42,   66,   99,  143,  200, ...
  1, 3,  9, 20,  42,  75,  132,  212,  333,  497, ...
  1, 4, 12, 30,  66, 132,  245,  429,  715, 1144, ...
  1, 4, 15, 40,  99, 212,  429,  800, 1430, 2424, ...
  1, 5, 18, 55, 143, 333,  715, 1430, 2700, 4862, ...
  1, 5, 22, 70, 200, 497, 1144, 2424, 4862, 9225, ...
  ...
Reading by antidiagonals, we get:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  3,  2,  1;
  1, 3,  5,  5,  3,   1;
  1, 3,  7,  8,  7,   3,   1;
  1, 4,  9, 14, 14,   9,   4,  1;
  1, 4, 12, 20, 25,  20,  12,  4,  1;
  1, 5, 15, 30, 42,  42,  30, 15,  5,  1;
  1, 5, 18, 40, 66,  75,  66, 40, 18,  5, 1;
  1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1;
  ...

Taylor Brysiewicz, Necklaces count polynomial parametric osculants, arXiv:1807.03408 [math.AG], 2018.
A. Elashvili, M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238. MR1691428 (2000c:13006).
A. Elashvili, M. Jibladze, D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188. MR1719140 (2000j:05009). See p. 174.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
I. M. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory, Ser. A, 64, 1993, 189-215, Theorem 9.4.
J. E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245, eq. (5).

This array is very similar to but different from A011847.
Cf. A051168, A092964, A241926, A047996, A037306, A245559.
Rows include A001840, A006918, A051170, A011796, A011797, A031164. Main diagonal is A022553.

# To produce the first 10 rows and columns (as on page 174 of the Elashvili et al. 1999 reference):
with(numtheory):
cnk:=(n,k) -> add(mobius(n/d)*d, d in divisors(gcd(n,k)));
anmk:=(n,m,k)->(1/(n+m))*add( cnk(d,k)*binomial((n+m)/d,n/d), d in divisors(gcd(n,m))); # anmk(n,m,k) is the value of a_k(n,m) as in Theorem 1, Equation (4), of the Elashvili et al. 1999 reference.
r2:=(n,k)->[seq(anmk(n,m,k),m=1..10)];
for n from 1 to 10 do lprint(r2(n,1)); od:

rows = 12;
cnk[n_, k_] := Sum[MoebiusMu[n/d] d, {d , Divisors[GCD[n, k]]}];
anmk[n_, m_, k_] := (1/(n+m)) Sum[cnk[d, k] Binomial[(n+m)/d, n/d], {d, Divisors[GCD[n, m]]}];
r2[n_, k_] := Table[anmk[n, m, k], {m, 1, rows}];
T = Table[r2[n, 1], {n, 1, rows}];
Table[T[[n-k+1, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 05 2018, from Maple *)

A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121

Offset: 0

Gus Wiseman, Apr 05 2019

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside it.

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)
        (31)   (41)    (42)     (52)      (53)       (63)        (64)
        (211)  (221)   (51)     (61)      (62)       (72)        (73)
               (311)   (222)    (511)     (71)       (81)        (82)
               (2111)  (411)    (2221)    (611)      (711)       (91)
                       (2211)   (4111)    (2222)     (6111)      (811)
                       (3111)   (22111)   (5111)     (22221)     (7111)
                       (21111)  (31111)   (22211)    (51111)     (22222)
                                (211111)  (41111)    (222111)    (61111)
                                          (221111)   (411111)    (222211)
                                          (311111)   (2211111)   (511111)
                                          (2111111)  (3111111)   (2221111)
                                                     (21111111)  (4111111)
                                                                 (22111111)
                                                                 (31111111)
                                                                 (211111111)

Colin Barker, Table of n, a(n) for n = 0..1000
N. Guru Sharan, Rook decomposition of the Partition function, arXiv:2507.20260 [math.CO], 2025. See p. 5.
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006918, A065770, A115994, A117485, A257990, A297113, A325135, A325166, A325169, A325170.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]==2&]],{n,0,30}]

concat([0,0,0], Vec(x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)) + O(x^80))) \\ Colin Barker, Apr 08 2019

From Colin Barker, Apr 08 2019: (Start)
G.f.: x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7.
a(n) = 2*n - 4 for n>4 and even.
a(n) = 2*n - 5 for n>4 and odd.
(End)

A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 5, 3, 1, 1, 1, 5, 8, 10, 3, 1, 1, 1, 6, 14, 26, 16, 4, 1, 1, 1, 7, 20, 61, 60, 29, 4, 1, 1, 1, 8, 30, 128, 243, 184, 45, 5, 1, 1, 1, 9, 40, 254, 800, 1228, 488, 75, 5, 1, 1, 1, 10, 55, 467, 2518, 7252, 6684, 1509, 115, 6, 1

Offset: 0

Andrew Howroyd, Apr 08 2020

T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333737.

Terms may be computed without generating each graph by enumerating the graphs by degree sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A192517 can be used to extend this method to the unlabeled case.

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1  1   1    1     1      1       1 ...
  1 | 1 1  1   1    1     1      1       1 ...
  2 | 1 2  3   4    5     6      7       8 ...
  3 | 1 2  5   8   14    20     30      40 ...
  4 | 1 3 10  26   61   128    254     467 ...
  5 | 1 3 16  60  243   800   2518    6999 ...
  6 | 1 4 29 184 1228  7252  38194  175369 ...
  7 | 1 4 45 488 6684 78063 772243 6254652 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals 0..26)

Rows n=0..4 are A000012, A000012, A000027(n+1), A006918(n+1), A333897.
Columns k=0..5 are A000012, A008619, A000990, A333894, A333895, A333896.
Cf. A188403, A192517, A263293, A333161, A333330, A333737, A334546.

A117486 Expansion of 1/((1-x)(1-x^2)(1-x^3)*(1-x^4))^2.

Original entry on oeis.org

1, 2, 5, 10, 20, 34, 59, 94, 149, 224, 334, 480, 685, 950, 1307, 1762, 2357, 3100, 4050, 5220, 6685, 8466, 10659, 13294, 16494, 20298, 24859, 30234, 36609, 44056, 52806, 62952, 74770, 88380, 104112, 122116, 142786, 166304, 193134, 223504, 257954, 296756, 340544

Offset: 0

Alford Arnold, Mar 22 2006

Molien series for S_4 X S_4, cf. A001400.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).
Index entries for two-way infinite sequences

Column four of table A115994.
Cf. A001400, A001399, A117485, A117487.
Cf. A000027, A006918, A115994.

n:=4; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H);

CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^4))^2,{x,0,50}],x] (* Harvey P. Dale, Jul 22 2012 *)

a(n):=floor(2*floor(-n/3)*cos(2*%pi*(n+1)/3)/81+(n+2)*cos(%pi*n/ 2)/128+(n+1)*(2835*(n^2+29*n+246)*(-1)^n+6*n^6+414*n^5+11556*n^4 +166944*n^3 +1320045*n^2+5489625*n+10008110)/ 17418240+1/2); /* Tani Akinari, Nov 14 2012 */

Vec(1 / ((1 - x)^8*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^40)) \\ Colin Barker, Apr 07 2019

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))^2.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) - 4*a(n-5) + 4*a(n-6) + 4*a(n-7) + 2*a(n-8) - 10*a(n-10) + 2*a(n-12) + 4*a(n-13) + 4*a(n-14) - 4*a(n-15) - a(n-16) - 2*a(n-17) + a(n-18) + 2*a(n-19) - a(n-20) for n>19. - Colin Barker, Apr 07 2019

Entry revised by N. J. A. Sloane, Mar 10 2007

A325165 Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4, 0, 2, 0, 0, 0, 5, 0, 3, 2, 0, 0, 0, 6, 0, 4, 4, 0, 0, 0, 0, 7, 0, 5, 6, 3, 0, 0, 0, 0, 8, 0, 7, 8, 6, 0, 0, 0, 0, 0, 9, 0, 9, 10, 9, 4, 0, 0, 0, 0, 0, 10, 0, 13, 12, 12, 8, 0, 0, 0, 0, 0, 0, 11

Offset: 0

Gus Wiseman, Apr 05 2019

The k-th part of the inner lining partition of an integer partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. For example, the partition (6,5,5,3) has diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
which has diagonal distances from the lower-right boundary equal to
  3 3 3 2 1 1
  3 2 2 2 1
  2 2 1 1 1
  1 1 1
so the inner lining sequence is (9,6,4) with last part 4, so (6,5,5,3) is counted under T(19,4).

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  0  3
  0  1  0  0  4
  0  2  0  0  0  5
  0  3  2  0  0  0  6
  0  4  4  0  0  0  0  7
  0  5  6  3  0  0  0  0  8
  0  7  8  6  0  0  0  0  0  9
  0  9 10  9  4  0  0  0  0  0 10
  0 13 12 12  8  0  0  0  0  0  0 11
  0 17 16 15 12  5  0  0  0  0  0  0 12
  0 24 20 18 16 10  0  0  0  0  0  0  0 13
  0 31 28 21 20 15  6  0  0  0  0  0  0  0 14
  0 42 36 27 24 20 12  0  0  0  0  0  0  0  0 15
  0 54 50 33 28 25 18  7  0  0  0  0  0  0  0  0 16
  0 71 64 45 32 30 24 14  0  0  0  0  0  0  0  0  0 17
  0 90 86 57 40 35 30 21  8  0  0  0  0  0  0  0  0  0 18
Row n = 9 counts the following partitions (empty columns not shown):
  (72)       (63)      (54)     (9)
  (333)      (522)     (432)    (81)
  (621)      (531)     (441)    (711)
  (5211)     (4221)    (3222)   (6111)
  (42111)    (4311)    (3321)   (51111)
  (321111)   (32211)   (22221)  (411111)
  (2211111)  (33111)            (3111111)
             (222111)           (21111111)
                                (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000041. Column k = 1 is A188674.

Cf. A006918, A115720, A115994, A117485, A252464, A257990, A325163, A325166, A325168.

pml[ptn_]:=If[ptn=={},{},FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,ptn][[-3]]];
Table[Length[Select[IntegerPartitions[n],Total[pml[#]]==k&]],{n,0,10},{k,0,n}]

T(n) = {my(v=Vec(1+sum(k=1, sqrtint(n), x^(k^2)/((1-y*x^k)*prod(j=1, k-1, 1 - x^j + O(x^(n+1-k^2))))^2))); vector(#v, i, Vecrev(v[i], -i))}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 19 2023

G.f.: A(x,y) = 1 + Sum_{k>=1} x^(k^2)/((1 - y*x^k) * Product_{j=1..k-1} (1 - x^j))^2. - Andrew Howroyd, Jan 19 2023

A144678 Related to enumeration of quantum states (see reference for precise definition).

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 16, 22, 28, 34, 40, 50, 60, 70, 80, 95, 110, 125, 140, 161, 182, 203, 224, 252, 280, 308, 336, 372, 408, 444, 480, 525, 570, 615, 660, 715, 770, 825, 880, 946, 1012, 1078, 1144, 1222, 1300, 1378, 1456, 1547, 1638, 1729, 1820, 1925, 2030, 2135

Offset: 0

N. J. A. Sloane, Feb 06 2009

The Gi2 triangle sums of the triangle A159797 are linear sums of shifted versions of the sequence given above, i.e., Gi2(n) = a(n-1) + 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). For the definitions of the Gi2 and other triangle sums see A180662. [Johannes W. Meijer, May 20 2011]
Partial sums of 1,1,1,1, 3,3,3,3, 6,6,6,6,..., the quadruplicated A000217. - R. J. Mathar, Aug 25 2013
Number of partitions of n into two different parts of size 4 and two different parts of size 1. a(4) = 7: 4, 4', 1111, 1111', 111'1', 11'1'1', 1'1'1'1'. - Alois P. Heinz, Dec 22 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=4]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).

Cf. A000292, A006918, A144677, A144679, A190718.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021

n:=80; lambda:=4; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b),snc];
od:
S10b;
A144678 := proc(n) option remember;
   local k;
   sum(A190718(n-k),k=0..3)
end:
A190718:= proc(n)
   binomial(floor(n/4)+3,3)
end:
seq(A144678(n),n=0..54); # Johannes W. Meijer, May 20 2011

a[n_] = (r = Mod[n, 4]; (4+n-r)(8+n-r)(3+n+2r)/96); Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Sep 02 2011 *)
LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1}, {1,2,3,4,7,10,13,16,22,28}, 60] (* G. C. Greubel, Oct 18 2021 *)

Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2013

def A144678_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^4))^2 ).list()
A144678_list(60) # G. C. Greubel, Oct 18 2021

From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A190718(n-3) + A190718(n-2) + A190718(n-1) + A190718(n).
a(n-3) + a(n-2) + a(n-1) + a(n) = A122046(n+3).
G.f.: 1/((x-1)^4*(x^3+x^2+x+1)^2). (End)
a(n) = A009531(n+5)/16 + (n+5)*(2*n^2+20*n+33+3*(-1)^n)/192 . - R. J. Mathar, Jun 20 2013
a(n) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - Wesley Ivan Hurt, Jul 21 2014
From Alois P. Heinz, Dec 22 2021: (Start)
G.f.: 1/((1-x)*(1-x^4))^2.
a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End)

cp's OEIS Frontend

A006584 If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12.

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A111384 a(n) = binomial(n,3) - binomial(floor(n/2),3) - binomial(ceiling(n/2),3).

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A144677 Related to enumeration of quantum states (see reference for precise definition).

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A325170 Heinz numbers of integer partitions with origin-to-boundary graph-distance equal to 2.

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A245558 Square array read by antidiagonals: T(n,k) = number of n-tuples of nonnegative integers (u_0,...,u_{n-1}) satisfying Sum_{j=0..n-1} j*u_j == 1 mod n and Sum_{j=0..n-1} u_j = m.

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A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

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A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

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A006584 If n mod 2 = 0 then n(n^2-4)/12 else n(n^2-1)/12.

A117486 Expansion of 1/((1-x)(1-x^2)(1-x^3)*(1-x^4))^2.