A000601 -id:A000601 - cp's OEIS frontend

A002622 Number of partitions of at most n into at most 5 parts.

1, 2, 4, 7, 12, 19, 29, 42, 60, 83, 113, 150, 197, 254, 324, 408, 509, 628, 769, 933, 1125, 1346, 1601, 1892, 2225, 2602, 3029, 3509, 4049, 4652, 5326, 6074, 6905, 7823, 8837, 9952, 11178, 12520, 13989, 15591, 17338, 19236, 21298, 23531, 25949, 28560, 31378, 34412

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 29*x^6 + 42*x^7 + 60*x^8 + ...
a(2) = 4 with partitions 0, 1, 2, 1+1. a(3) = 7 with partitions 0, 1, 2, 1+1, 3, 2+1, 1+1+1. - _Michael Somos_, Apr 24 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 0, 2, -1).

Cf. A001401 (first differences). Column 5 of A092905.

CoefficientList[Series[1/((1 - x)^2 (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Apr 25 2014 *)
LinearRecurrence[{2, 0, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 0, 2, -1},  {1, 2, 4, 7, 12, 19, 29, 42, 60, 83, 113, 150, 197, 254, 324, 408},  48] (* Georg Fischer, Feb 27 2019 *)

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 5, 1-x^i))) \\ Altug Alkan, Mar 30 2018

G.f.: 1/[(1+x^2)*(1-x^3)*(1-x)^4*(1-x^5)*(1+x)^2]. (Corrected Mar 31 2018)
a(n)= 2*a(n-1) -a(n-3) -a(n-5) +2*a(n-8) -a(n-11) -a(n-13) +2*a(n-15) -a(n-16).
G.f.: 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^3) * (1 - x^4) * (1 - x^5)). - Michael Somos, Apr 24 2014
Euler transform of length 5 sequence [ 2, 1, 1, 1, 1]. - Michael Somos, Apr 24 2014
a(n) = a(n-1) + A001401(n). - Michael Somos, Apr 24 2014
a(n) = round((n+1)*(6*n^4+234*n^3+3326*n^2+20674*n+50651+675*(-1)^n)/86400). - Tani Akinari, May 05 2014

A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 4, 7, 19, 32, 68, 114, 210, 336, 562, 862, 1349, 1987, 2950, 4201, 5991, 8278, 11422, 15386, 20660, 27218, 35718, 46158, 59401, 75475, 95494, 119545, 149035, 184118, 226562, 276620, 336470, 406490, 489344, 585572, 698397, 828549, 979896

Offset: 0

N. J. A. Sloane

Also the number of ways in which to label the vertices of the cube (or faces of the octahedron) with nonnegative integers summing to n, where labelings that differ only by rotation or reflection are considered the same. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 26 2004

			Representatives of the seven classes of 3 X 3 binary matrices are:
[ 1 1 1 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 1 ] [ 0 1 1 ] [ 0 1 1 ] [ 0 1 1 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 0 ] [ 1 0 0 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 1 1 ] [ 1 0 0 ].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1052.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,2,-5,7,-4,2,2,-8,10,-8,2,2,-4,7,-5,2,-1,-2,3,-1).
Index entries for sequences related to binary matrices
Index entries for two-way infinite sequences

Column k=3 of A363349.

Cf. A000601, A005232, A006382, A006380, A002727, A006148.

Vec((1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48 + O(x^41)) \\ Andrew Howroyd, May 30 2023

G.f.: (1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48.

G.f.: (x^14 - 2*x^13 + 3*x^12 - 2*x^11 + 5*x^10 - 4*x^9 + 7*x^8 - 4*x^7 + 7*x^6 - 4*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 2*x + 1)/(x^6 - 1)/(x^2 + 1)^2/(x^2 + x + 1)/(x + 1)^3/(x - 1)^7.

Entry revised by Vladeta Jovovic, Aug 05 2000

Definition corrected by Max Alekseyev, Feb 05 2010

A006380 Number of equivalence classes of 4 X n binary matrices when one can permute rows, permute columns and complement columns.

Original entry on oeis.org

1, 3, 8, 19, 41, 81, 153, 273, 468, 774, 1240, 1930, 2933, 4356, 6341, 9064, 12743, 17643, 24093, 32479, 43270, 57019, 74377, 96103, 123089, 156354, 197081, 246622, 306519, 378520, 464614, 567028, 688276, 831169, 998845, 1194793, 1422899, 1687447, 1993182

Offset: 0

N. J. A. Sloane

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Index entries for sequences related to binary matrices
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-2,2,5,-8,6,-8,5,2,-2,2,-5,4,-1).
Index entries for two-way infinite sequences

Row n=4 of A363349.

Cf. A000601, A006148, A006383.

LinearRecurrence[{4,-5,2,-2,2,5,-8,6,-8,5,2,-2,2,-5,4,-1},{1,3,8,19,41,81,153,273,468,774,1240,1930,2933,4356,6341,9064},40] (* Harvey P. Dale, Nov 23 2024 *)

Vec((1 - x + x^2 + x^4 + x^6 - x^7 + x^8)/((1 - x)^8*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2) + O(x^41)) \\ Andrew Howroyd, May 30 2023

G.f.: (1 - x + x^2 + x^4 + x^6 - x^7 + x^8)/((1 - x)^8*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2). - Andrew Howroyd, May 30 2023

Terms a(7) onwards from Max Alekseyev, Feb 05 2010

A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 3, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 6, 11, 19, 10, 4, 1, 1, 1, 7, 16, 41, 32, 16, 4, 1, 1, 1, 8, 23, 81, 101, 68, 20, 5, 1, 1, 1, 9, 31, 153, 299, 301, 114, 29, 5, 1, 1, 1, 10, 41, 273, 849, 1358, 757, 210, 35, 6, 1

Offset: 0

Andrew Howroyd, May 28 2023

T(n,k) is also the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows and columns.

			Array begins:
======================================================
n/k| 0 1  2   3    4     5      6       7        8 ...
---+--------------------------------------------------
0  | 1 1  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        9 ...
3  | 1 2  4   7   11    16     23      31       41 ...
4  | 1 3  8  19   41    81    153     273      468 ...
5  | 1 3 10  32  101   299    849    2290     5901 ...
6  | 1 4 16  68  301  1358   6128   27008   114763 ...
7  | 1 4 20 114  757  5567  43534  343656  2645494 ...
8  | 1 5 29 210 1981 23350 319119 4633380 67013431 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

A259344 is the same array without the first row and column read by upward antidiagonals.
Columns k=0..6 are A000012, A004526(n+2), A005232, A006381, A006382, A056204, A056205.
Rows n=2..4 are A000027(n+1), A000601, A006380.
Main diagonal is A006383.
Cf. A028657, A241956, A362905.

\\ Compare A028657.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
T(n, k)={if(n==0, 1, my(s=0); forpart(q=n, my(e=1<

A092905 Triangle, read by rows, such that the partial sums of the n-th row form the n-th diagonal, for n>=0, where each row begins with 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 6, 4, 2, 1, 1, 6, 9, 7, 4, 2, 1, 1, 7, 12, 11, 7, 4, 2, 1, 1, 8, 16, 16, 12, 7, 4, 2, 1, 1, 9, 20, 23, 18, 12, 7, 4, 2, 1, 1, 10, 25, 31, 27, 19, 12, 7, 4, 2, 1, 1, 11, 30, 41, 38, 29, 19, 12, 7, 4, 2, 1, 1, 12, 36, 53, 53, 42, 30, 19, 12, 7, 4, 2, 1

Offset: 0

Paul D. Hanna, Mar 12 2004

Row sums form A000070, which is the partial sums of the partition numbers (A000041). Rows read backwards converge to the row sums (A000070).
From Alford Arnold, Feb 07 2010: (Start)
The table can also be generated by summing sequences embedded within Table A008284
For example,
1 1 1 1 ... yields 1 2 3 4 ...
1 1 2 2 3 3 ... yields 1 2 4 6 9 12 ...
1 1 2 3 4 5 7 ... yields 1 2 4 7 11 16 ...
(End)
T(n,k) is also count of all 'replacable' cells in the (Ferrers plots of) the partitions on n in exactly k parts. [Wouter Meeussen, Sep 16 2010]
From Wolfdieter Lang, Dec 03 2012: (Start)
The triangle entry T(n,k) is obtained from triangle A072233 by summing the entries of column k up to n (see the partial sum type o.g.f. given by Vladeta Jovovic in the formula section).
  Therefore, the  o.g.f. for the sequence in column k is x^k/((1-x)* product(1-x^j,j=1..k)).
The triangle with entry a(n,m) = T(n-1,m-1), n >= 1, m = 1, ..., n, is obtained from the partition array A103921 when in row n all entries belonging to part number m are summed (a conjecture). (End)

			The fourth row (n=3) is {1,3,2,1} and the fourth diagonal is the partial sums of the fourth row: {1,4,6,7,7,7,7,7,...}.
The triangle T(n,k) begins:
n\k 0  1  2  3  4  5  6  7  8  9 10 11 12  ...
0   1
1   1  1
2   1  2  1
3   1  3  2  1
4   1  4  4  2  1
5   1  5  6  4  2  1
6   1  6  9  7  4  2  1
7   1  7 12 11  7  4  2  1
8   1  8 16 16 12  7  4  2  1
9   1  9 20 23 18 12  7  4  2  1
10  1 10 25 31 27 19 12  7  4  2  1
11  1 11 30 41 38 29 19 12  7  4  2  1
12  1 12 36 53 53 42 30 19 12  7  4  2  1
... Reformatted by _Wolfdieter Lang_, Dec 03 2012
T(5,3)=4 because the partitions of 5 in exactly 3 parts are 221 and 311, and they give rise to partitions of 4 in four ways: 221->22 and 211, 311->211 and 31, since both their Ferrers plots have 2 'mobile cells' each. [_Wouter Meeussen_, Sep 16 2010]
T(5,3) = a(6,4) = 4 because the partitions of 6 with 4 parts are 1113 and 1122, with the number of distinct parts 2 and 2, respectively, summing to 4 (see the array A103921). An example for the conjecture given as comment above. - _Wolfdieter Lang_, Dec 03 2012

V. V. Kruchinin, The number of partitions of a natural number n into parts each of which is not less than m, Math. Notes 86 (4) (2009) 505-509
R. J. Mathar, Size of the set of residues of integer powers of fixed exponent, (2017), Table 11.

Antidiagonal sums form the partition numbers (A000041).
Cf. A000070.
Cf. A008284. [Alford Arnold, Feb 07 2010]
Columns: A087811, A000601, A002621, A002622, A288341 - A288345.

Maple
```
T(n,k)=if(n
				
```

(*Needs["DiscreteMath`Combinatorica`"]; partitionexact[n_, m_] := TransposePartition /@ (Prepend[ #1, m] & ) /@ Partitions[n - m, m] *); mobile[p_?PartitionQ]:=1+Count[Drop[p,-1]-Rest[p],?Positive]; Table[Tr[mobile/@partitionexact[n,k]],{n,12},{k,n}] (* _Wouter Meeussen, Sep 16 2010 *)

T(n, k) = sum_{j=0..k} T(n-k, j), with T(n, 0) = 1 for all n>=0. A000070(n) = sum_{k=0..n} T(n, k).

O.g.f.: (1/(1-y))*(1/Product(1-x*y^k, k=1..infinity)). - Vladeta Jovovic, Jan 29 2005

Several corrections by Wolfdieter Lang, Dec 03 2012

A002625 Expansion of 1/((1-x)^3(1-x^2)^2(1-x^3)).

Original entry on oeis.org

1, 3, 8, 17, 33, 58, 97, 153, 233, 342, 489, 681, 930, 1245, 1641, 2130, 2730, 3456, 4330, 5370, 6602, 8048, 9738, 11698, 13963, 16563, 19538, 22923, 26763, 31098, 35979, 41451, 47571, 54390, 61971, 70371, 79660, 89901, 101171, 113540, 127092, 141904, 158068, 175668, 194804, 215568

Offset: 0

N. J. A. Sloane

Number of (integer) partitions of n into 3 sorts of 1's, 2 sorts of 2's, and 1 sort of 3's. - Joerg Arndt, May 17 2013

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 205
Gerzson Keri and Patric R. J. Östergård, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,2,2,2,-4,-1,3,-1).

Partial sums of A097701.

A002625:=1/(z**2+z+1)/(z+1)**2/(z-1)**6; [Simon Plouffe in his 1992 dissertation.]

CoefficientList[Series[1/((1-x)^3*(1-x^2)^2*(1-x^3)),{x,0,50}],x] (* Vincenzo Librandi, Feb 25 2012 *)
LinearRecurrence[{3,-1,-4,2,2,2,-4,-1,3,-1},{1,3,8,17,33,58,97,153,233,342},50] (* Harvey P. Dale, Sep 24 2022 *)

Vec(1/(1-x)^3/(1-x^2)^2/(1-x^3)+O(x^99)) \\ Charles R Greathouse IV, Apr 30 2012

a(n) = floor((n+1)*(135*(-1)^n + 6*n^4 + 144*n^3 + 1256*n^2 + 4744*n + 6785)/8640+1/2). - Tani Akinari, Oct 07 2012

A181120 Partial sums of round(n^2/12) (A069905).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 11, 16, 23, 31, 41, 53, 67, 83, 102, 123, 147, 174, 204, 237, 274, 314, 358, 406, 458, 514, 575, 640, 710, 785, 865, 950, 1041, 1137, 1239, 1347, 1461, 1581, 1708, 1841, 1981, 2128, 2282, 2443, 2612, 2788, 2972, 3164, 3364, 3572

Offset: 0

Mircea Merca, Oct 04 2010

Number of triples of positive integers (a, b, c) such that 1 <= a <= b <= c and a + b + c <= n. - Leonhard Vogt, Apr 27 2017

			a(5) = 4 = 0 + 0 + 0 + 1 + 1 + 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..870
D. Barrera, M. J. Ibáñez, and S. Remogna, On the construction of trivariate near-best quasi-interpolants based on C^2 quartic splines on type-6 tetrahedral partitions, Journal of Computational and Applied, 2016, Volume 311, February 2017, Pages 252-261.
J. Brandts and A. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
Jan Brandts and Apo Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Partial sums of A069905.

a:= n-> round(1/(72)*(2*n^(3)+3*n^(2)-6*n)): seq(a(n), n=0..50);

a(n)=round(n*(2*n^2+3*n-6)/72) \\ Charles R Greathouse IV, May 23 2013

a(n) = round((2*n^3 + 3*n^2 - 6*n)/72).
a(n) = round((4*n^3 + 6*n^2 - 12*n - 7)/144).
a(n) = floor((2*n^3 + 3*n^2 - 6*n + 9)/72).
a(n) = ceiling((2*n^3 + 3*n^2 - 6*n + 9 - 16)/72).
a(n) = a(n-6) + (n^2 - 5*n + 8)/2, n > 5.
From R. J. Mathar, Oct 06 2010: (Start)
a(n) = (-1)^n/16 + n^3/36 - n^2/24 - n/12 + 7/144 - A049347(n)/9.
G.f.: x^4 / ( (1+x)*(1+x+x^2)*(x-1)^4 ). (End)
a(n) = A000601(n-3). - R. J. Mathar, Oct 11 2017

A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 0, 2, 1, 4, 2, 7, 4, 11, 7, 16, 11, 23, 16, 31, 23, 41, 31, 53, 41, 67, 53, 83, 67, 102, 83, 123, 102, 147, 123, 174, 147, 204, 174, 237, 204, 274, 237, 314, 274, 358, 314, 406, 358, 458, 406, 514, 458, 575, 514, 640, 575, 710, 640, 785, 710, 865, 785, 950

Offset: 0

Michael Somos, Feb 02 2015

Partitions of n into parts of size 3 and size 4 and two kinds of parts of size 2.
The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, u+v >= x+w, and x+u+v+w is even.
Euler transform of length 4 sequence [ 0, 2, 1, 1].

			G.f. = 1 + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + 7*x^6 + 4*x^7 + 11*x^8 + 7*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

Cf. A001400, A002620, A005044, A115264, A165188.

Cf. A241526, A000601, A181120.

I:=[1,0,2,1,4,2,7,4,11,7,16]; [n le 11 select I[n] else 2*Self(n-2)+Self(n-3)-2*Self(n-5)-2*Self(n-6)+Self(n-8)+2*Self(n-9)-Self(n-11): n in [1..60]]; // Vincenzo Librandi, Feb 03 2015

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 12 n^2 + 33 n + 54, 21 n^2 + 132 n + 288], 288];
a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -11 - n]; s SeriesCoefficient[ 1 / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, u + v >= x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];
CoefficientList[Series[1 / (1 - 2 x^2 - x^3 + 2 x^5 + 2 x^6 - x^8 - 2 x^9 + x^11), {x, 0, 60}], x] (* Vincenzo Librandi, Feb 03 2015 *)

{a(n) = (n^3 + if(n%2, 12*n^2 + 33*n + 54, 21*n^2 + 132*n + 288)) \ 288};

{a(n) = my(s=1); if( n<0, s=-1; n=-11-n); s * polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

G.f.: 1 / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
a(n) = -a(-11-n) for all n in Z.
a(n+3) - a(n) = 0 if n even else floor((n+7)^2 / 16).
0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
a(n) - a(n-2) = A005044(n+3) for all n in Z.
a(n) + a(n-1) = A001400(n) for all n in Z.
a(n) + a(n-2) = A165188(n+1) for all n in Z.
a(n) = A115264(n) - A115264(n-1) for all n in Z.
a(2*n) - a(2*n-6) = a(2*n+3) - a(2*n-3) = A002620(n+2) for all n in Z. - Michael Somos, Feb 11 2015
a(n) = (2*n^3+33*n^2+181*n+234+3*(3*n^2+33*n+86)*(-1)^n+84*(-1)^((2*n+1-(-1)^n)/4)-96*((1+(-1)^n)*floor(((2*n+9+(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24))+(1-(-1)^n)*floor(((2*n+5+(-1)^n-6*(-1)^((2*n-1+(-1)^n)/4))/24))))/576. - Luce ETIENNE, May 22 2015

A002626 Expansion of 1/((1-x)^3 (1-x^2)^2 (1-x^3) (1-x^4)).

Original entry on oeis.org

1, 3, 8, 17, 34, 61, 105, 170, 267, 403, 594, 851, 1197, 1648, 2235, 2981, 3927, 5104, 6565, 8351, 10529, 13152, 16303, 20049, 24492, 29715, 35841, 42972, 51255, 60813, 71820, 84423, 98826, 115203, 133791, 154794, 178486, 205104, 234962, 268334, 305578

Offset: 0

N. J. A. Sloane

E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 206
Index entries for linear recurrences with constant coefficients, signature (3, -1, -4, 3, -1, 3, 0, -3, 1, -3, 4, 1, -3, 1).

LinearRecurrence[{3, -1, -4, 3, -1, 3, 0, -3, 1, -3, 4, 1, -3, 1}, {1, 3, 8, 17, 34, 61, 105, 170, 267, 403, 594, 851, 1197, 1648}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

Vec(1/(1-x)^3/(1-x^2)^2/(1-x^3)/(1-x^4)+O(x^99)) \\ Charles R Greathouse IV, Apr 30 2012

a(n) = floor((n+1)*(n+13)*(135*(-1)^n + 2*n^4 + 56*n^3 + 570*n^2 + 2492*n + 4175)/69120 + 1/2). - Tani Akinari, Nov 07 2012

A139672 Convolution of A008619 and A001400.

Original entry on oeis.org

1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191, 257, 346, 451, 587, 746, 946, 1177, 1461, 1786, 2178, 2623, 3151, 3746, 4443, 5223, 6126, 7131, 8283, 9558, 11007, 12603, 14403, 16377, 18588, 21003, 23692, 26618, 29858, 33372, 37244, 41430, 46022, 50972

Offset: 1

Alford Arnold, Apr 29 2008, May 01 2008

nonn

This is row 21 of a table of values related to Molien series. It is the product of the sequence on row 3 (A008619) with the sequence on row 7 (A001400).
This table may be constructed by moving the rows of table A008284 to prime locations and generating the composite locations by multiplication in a manner similar to the calculation illustrated in the present sequence.
Rows 1 thru 20 and 22 thru 25 are as follows:
A000012 A008619 A000027 A001399 A002620 A001400 A000217 A006918 A000601 A001401 A002623 A001402 A002621 A097701 A000292 A008630 A002624 A008631 A057524 A002622 A008632 A001752 A117485.

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

a:= proc(n) local m, r; m:= iquo (n, 12, 'r'); r:= r+1; (19+ (145+ (260+ 15* (r+9)*r+ (405+ 90*r+ 216*m) *m) *m) *m) *m/5+ [0, 1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191][r]+ [0, 16, 37, 77, 128, 208, 307, 447, 616, 840, 1105, 1441][r]*m/2+ [0, 52, 119, 213, 328, 476, 651, 865, 1112, 1404, 1735, 2117][r]*m^2/2 end: seq (a(n), n=1..50); # Alois P. Heinz, Nov 10 2008

CoefficientList[Series[x/((x^2+x+1)(x^2+1)(x+1)^3 (x-1)^6),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,-3,0,-1,2,2,-1,0,-3,1,2,-1},{0,1,2,5,9,17,27,44,65,97,136,191,257},50] (* Harvey P. Dale, Feb 17 2016 *)

G.f.: x/((x^2+x+1)*(x^2+1)*(x+1)^3*(x-1)^6). - Alois P. Heinz, Nov 10 2008

a(n)= -A049347(n)/27 +(2*n+11)*(6*n^4+132*n^3+914*n^2+2068*n+1055)/69120 -(-1)^n*(51/512+n^2/256+11*n/256+A057077(n)/32 ). - R. J. Mathar, Nov 21 2008

More terms from Alois P. Heinz, Nov 10 2008

Corrected A-number in definition. Added formula. - R. J. Mathar, Nov 21 2008

cp's OEIS Frontend

A002622 Number of partitions of at most n into at most 5 parts.

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A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

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A006380 Number of equivalence classes of 4 X n binary matrices when one can permute rows, permute columns and complement columns.

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A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

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A092905 Triangle, read by rows, such that the partial sums of the n-th row form the n-th diagonal, for n>=0, where each row begins with 1.

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A002625 Expansion of 1/((1-x)^3*(1-x^2)^2*(1-x^3)).

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A181120 Partial sums of round(n^2/12) (A069905).

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A002625 Expansion of 1/((1-x)^3(1-x^2)^2(1-x^3)).