A006002 -id:A006002 - cp's OEIS frontend

A163274 a(n) = n^4*(n+1)^2/2.

0, 2, 72, 648, 3200, 11250, 31752, 76832, 165888, 328050, 605000, 1054152, 1752192, 2798978, 4321800, 6480000, 9469952, 13530402, 18948168, 26064200, 35280000, 47064402, 61960712, 80594208, 103680000, 132031250, 166567752, 208324872

Offset: 0

Omar E. Pol, Jul 24 2009

Row sums of triangle A163284.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006002, A099903, A163102, A163275, A163276, A163277, A163284.

Table[(n^4 (n+1)^2)/2,{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,2,72,648,3200,11250,31752},30] (* Harvey P. Dale, May 07 2012 *)

a(n)=n^4*(n+1)^2/2 \\ Charles R Greathouse IV, Oct 07 2015

From R. J. Mathar, Jul 29 2009: (Start)
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: -2*x*(1 + 29*x + 93*x^2 + 53*x^3 + 4*x^4)/(x-1)^7. (End)
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi^2/3 + Pi^4/45 - 4*zeta(3) - 10.
Sum_{n>=1} (-1)^(n+1)/a(n) = 10 + Pi^2/3 + 7*Pi^4/360 - 16*log(2) - 3*zeta(3). (End)

More terms from R. J. Mathar, Jul 29 2009

A163275 a(n) = n^5*(n+1)^2/2.

Original entry on oeis.org

0, 2, 144, 1944, 12800, 56250, 190512, 537824, 1327104, 2952450, 6050000, 11595672, 21026304, 36386714, 60505200, 97200000, 151519232, 230016834, 341067024, 495219800, 705600000, 988352442, 1363135664, 1853666784

Offset: 0

Omar E. Pol, Jul 24 2009

Row sums of triangle A163285.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A006002, A099903, A163102, A163274, A163276, A163277, A163285.

A163275 := proc(n) n^5*(n+1)^2/2 ; end proc: seq(A163275(n),n=0..60) ; # R. J. Mathar, Feb 05 2010

Table[(1/2)*n^5*(n + 1)^2, {n,0,50}] (* or *) LinearRecurrence[{8,-28,56, -70,56,-28,8,-1}, {0,2,144,1944,12800,56250,190512,537824}, 50] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*x*(1+64*x+424*x^2+584*x^3+179*x^4+8*x^5)/(x-1)^8 + O(x^50))) \\ G. C. Greubel, Dec 12 2016

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
G.f.: 2*x*(1 + 64*x + 424*x^2 + 584*x^3 + 179*x^4  +8*x^5)/(x-1)^8. (End)
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 12 -5*Pi^2/3 - 2*Pi^4/45 + 6*zeta(3) + 2*zeta(5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 20*log(2) + 9*zeta(3)/2 + 15*zeta(5)/8 - 12 - Pi^2/2 - 7*Pi^4/180. (End)

Extended by R. J. Mathar, Feb 05 2010

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

2, 9, 5, 24, 18, 8, 50, 42, 27, 11, 90, 80, 60, 36, 14, 147, 135, 110, 78, 45, 17, 224, 210, 180, 140, 96, 54, 20, 324, 308, 273, 225, 170, 114, 63, 23, 450, 432, 392, 336, 270, 200, 132, 72, 26, 605, 585, 540, 476, 399, 315

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213820.
Antidiagonal sums: A153978.
Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.
Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.
Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
2....9....24....50....90....147
5....18...42....80....135...210
8....27...60....110...180...273
11...36...78....140...225...336
14...45...96....170...270...399
17...54...114...200...315...462

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A212500

b[n_]:=n;c[n_]:=3n-1;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213819 *)
Table[t[n,n],{n,1,40}] (* A213820 *)
d/2 (* A002414 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A153978 *)
s1/2 (* A001296 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.

A267370 Partial sums of A140091.

Original entry on oeis.org

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510

Offset: 0

Bruno Berselli, Jan 13 2016

After 0, this sequence is the third column of the array in A185874.

Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

			The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005581, A051744, A105163, A140091, A185874.

Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).

Magma
```
[n*(n+1)*(n+5)/2: n in [0..50]];
```

Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* Harvey P. Dale, Jul 18 2019 *)

PARI
```
vector(50, n, n--; n*(n+1)*(n+5)/2)
```
Sage
```
[n*(n+1)*(n+5)/2 for n in (0..50)]
```

O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

A317303 Numbers k such that both Dyck paths of the symmetric representation of sigma(k) have a central peak.

Original entry on oeis.org

2, 7, 8, 9, 16, 17, 18, 19, 20, 29, 30, 31, 32, 33, 34, 35, 46, 47, 48, 49, 50, 51, 52, 53, 54, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 154, 155, 156, 157, 158, 159, 160

Offset: 1

Omar E. Pol, Aug 27 2018

Also triangle read by rows which gives the odd-indexed rows of triangle A014132.
There are no triangular number (A000217) in this sequence.
For more information about the symmetric representation of sigma see A237593 and its related sequences.
Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an odd number of peaks. - Omar E. Pol, Sep 13 2018

			Written as an irregular triangle in which the row lengths are the odd numbers, the sequence begins:
    2;
    7,   8,   9;
   16,  17,  18,  19,  20;
   29,  30,  31,  32,  33,  34,  35;
   46,  47,  48,  49,  50,  51,  52,  53,  54;
   67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77;
   92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104;
  121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135;
...
Illustration of initial terms:
-----------------------------------------------------------
   k   sigma(k)   Diagram of the symmetry of sigma
-----------------------------------------------------------
                    _         _ _ _             _ _ _ _ _
                  _| |       | | | |           | | | | | |
   2      3      |_ _|       | | | |           | | | | | |
                             | | | |           | | | | | |
                            _|_| | |           | | | | | |
                          _|  _ _|_|           | | | | | |
                  _ _ _ _|  _| |               | | | | | |
   7      8      |_ _ _ _| |_ _|               | | | | | |
   8     15      |_ _ _ _ _|              _ _ _| | | | | |
   9     13      |_ _ _ _ _|             |  _ _ _|_| | | |
                                        _| |    _ _ _|_| |
                                      _|  _|   |  _ _ _ _|
                                  _ _|  _|  _ _| |
                                 |  _ _|  _|    _|
                                 | |     |     |
                  _ _ _ _ _ _ _ _| |  _ _|  _ _|
  16     31      |_ _ _ _ _ _ _ _ _| |  _ _|
  17     18      |_ _ _ _ _ _ _ _ _| | |
  18     39      |_ _ _ _ _ _ _ _ _ _| |
  19     20      |_ _ _ _ _ _ _ _ _ _| |
  20     42      |_ _ _ _ _ _ _ _ _ _ _|
.
For the first nine terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central peak.
Compare with A317304.

Column 1 gives A130883, n >= 1.
Column 2 gives A033816, n >= 1.
Row sums give the odd-indexed terms of A006002.
Right border gives the positive terms of A014107, also the odd-indexed terms of A000096.
The union of A000217, A317304 and this sequence gives A001477.
Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317304. See also A317306.
Cf. A000203, A005408, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.

A330298 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 2 even numbers.

Original entry on oeis.org

0, 0, 0, 0, 2, 3, 9, 12, 24, 30, 50, 60, 90, 105, 147, 168, 224, 252, 324, 360, 450, 495, 605, 660, 792, 858, 1014, 1092, 1274, 1365, 1575, 1680, 1920, 2040, 2312, 2448, 2754, 2907, 3249, 3420, 3800, 3990, 4410, 4620, 5082, 5313, 5819, 6072, 6624, 6900, 7500, 7800, 8450, 8775, 9477

Offset: 0

Enrique Navarrete, Feb 29 2020

The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).

			For n=6, a(6) = 9 and the 9 subsets are: {1,2,4}, {1,2,6}, {1,4,6}, {2,3,4}, {2,3,6}, {2,4,5}, {2,5,6}, {3,4,6}, {4,5,6}.

Colin Barker, Table of n, a(n) for n = 0..1000
J.S. Seneschal, Oblong Prism Illustration
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A330299, A330300.
Cf. A000217, A002378.
Interleaves the positive integers of A006002 and A027480.

a[n_] := Ceiling[n/2] * Binomial[Floor[n/2], 2]; Array[a, 55, 0] (* Amiram Eldar, Mar 01 2020 *)
Table[Length[Select[Subsets[Range[n],{3}],Total[Boole[OddQ[#]]]==1&]],{n,0,60}] (* Harvey P. Dale, Jul 26 2020 *)

a(n) = ceil(n/2) * binomial(floor(n/2), 2) \\ Andrew Howroyd, Mar 01 2020

concat([0,0,0,0], Vec(x^4*(2 + x) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, Mar 02 2020

a(n) = ceiling(n/2) * binomial(floor(n/2), 2).
From Colin Barker, Mar 01 2020: (Start)
G.f.: x^4*(2 + x) / ((1 - x)^4*(1 + x)^3).
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>6. (End)
E.g.f.: (x*(-3 + x + x^2)*cosh(x) + (3 - x + x^3)*sinh(x))/16. - Stefano Spezia, Mar 02 2020
a(n) = 3/32-5*n^2/32-n/32+n^3/16+(-)^n*(n-3+n^2)/32. - R. J. Mathar, Mar 31 2025

A082146 Expansion of g.f.: (1+x^5)/((1-x^2)(1-x^3)(1-x^4)*(1-x^6)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 8, 9, 13, 12, 17, 18, 22, 24, 30, 30, 38, 40, 46, 50, 59, 60, 71, 75, 84, 90, 102, 105, 120, 126, 138, 147, 163, 168, 187, 196, 212, 224, 244, 252, 276, 288, 308, 324, 349, 360, 389, 405, 430, 450, 480, 495, 530, 550, 580, 605, 641, 660, 701, 726

Offset: 0

N. J. A. Sloane, Dec 30 2003

nonn

Poincaré series [or Poincare series] (or Molien series) for (P[x_0,x_1] ⊗ P[x_0,x_1])^(S_2).

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 199.

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

Cf. A089599, A091434, A091726, A091769.

Cf. A010875 (n mod 6). Contains A006002 and A212683. - Luce ETIENNE, Aug 14 2018

R:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (1-x^10)/(&*[1-x^j: j in [2..6]]) )); // G. C. Greubel, Apr 02 2023

seq(coeff(series((1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)), x,n+1),x,n),n=0..70); # Muniru A Asiru, Aug 15 2018

CoefficientList[Series[(1-x^10)/Product[1-x^(j+1), {j,5}], {x,0,70}], x] (* G. C. Greubel, Apr 02 2023 *)

Vec((1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)) + O(x^100)) \\ Michel Marcus, Mar 19 2014

def A082146_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^10)/prod(1-x^j for j in range(2,7)) ).list()
A082146_list(70) # G. C. Greubel, Apr 02 2023

a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-11) + a(n-13) - a(n-14).
G.f.: ( 1+x^2+x^4-x-x^3 ) / ( (1+x^2)*(1-x+x^2)*(1+x)^2*(1+x+x^2)^2*(1-x)^4 ). - R. J. Mathar, Oct 11 2011
a(n) = (120*floor(n/6)^3 + 60*(m+5)*floor(n/6)^2 - 20*(m^5-13*m^4 +60*m^3-116*m^2+74*m-18)*floor(n/6) - (19*m^5-245*m^4+1125*m^3-2185*m^2+1496*m-210) + (m^5-15*m^4+75*m^3-135*m^2+44*m+30)*(-1)^floor(n/6))/240 where m = (n mod 6). - Luce ETIENNE, Aug 14 2018

A163276 a(n) = n^6*(n+1)^2/2.

Original entry on oeis.org

0, 2, 288, 5832, 51200, 281250, 1143072, 3764768, 10616832, 26572050, 60500000, 127552392, 252315648, 473027282, 847072800, 1458000000, 2424307712, 3910286178, 6139206432, 9409176200, 14112000000, 20755401282, 29988984608

Offset: 0

Omar E. Pol, Jul 24 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A006002, A099903, A163102, A163274, A163275, A163277.

[n^6*(n+1)^2/2: n in [0..30]]; // Vincenzo Librandi, Dec 13 2016

seq((1/2)*n^6*(n+1)^2, n = 0 .. 25); # Emeric Deutsch, Aug 01 2009

Table[(1/2)*n^6*(n + 1)^2, {n,0,50}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1}, {0, 2, 288, 5832, 51200, 281250, 1143072, 3764768, 10616832}, 50] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*x*(1 + 135*x +1656*x^2 +4456*x^3 +3231*x^4 +585*x^5 +16*x^6)/(1-x)^9 + O(x^50))) \\ G. C. Greubel, Dec 12 2016

G.f.: 2*x*(1+135*x+1656*x^2+4456*x^3+3231*x^4+585*x^5+16*x^6)/(1-x)^9. - Colin Barker, May 05 2012
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi^2 + Pi^4/15 + 2*Pi^6/945 - 14 - 8*zeta(3) - 4*zeta(5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 14 + 2*Pi^2/3 + 7*Pi^4/120 + 31*Pi^6/15120 - 24*log(2) - 6*zeta(3) - 15*zeta(5)/4. (End)

Extended by Emeric Deutsch, Aug 01 2009

A163277 a(n) = n^7*(n+1)^2/2.

Original entry on oeis.org

0, 2, 576, 17496, 204800, 1406250, 6858432, 26353376, 84934656, 239148450, 605000000, 1403076312, 3027787776, 6149354666, 11859019200, 21870000000, 38788923392, 66474865026, 110505715776, 178774347800, 282240000000

Offset: 0

Omar E. Pol, Jul 24 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A006002, A099903, A163102, A163274, A163275, A163276.

[n^7*(n+1)^2/2: n in [0..30]]; // Vincenzo Librandi, Dec 13 2016

A163277 := proc(n) n^7*(n+1)^2/2 ; end proc: seq(A163277(n),n=0..60) ; \\ R. J. Mathar, Feb 05 2010

Table[(1/2)*n^7*(n + 1)^2, {n,0,50}] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*x*(1 +278*x +5913*x^2 +27760*x^3 +38435*x^4 +16434*x^5 +1867*x^6 +32*x^7)/(x-1)^10 + O(x^50))) \\ G. C. Greubel, Dec 12 2016

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = n^2*A163275(n).
G.f.: 2*x*(1 +278*x +5913*x^2 +27760*x^3 +38435*x^4 +16434*x^5 +1867*x^6 +32*x^7)/(x-1)^10. (End)
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 16 - 7*Pi^2/3 - 4*Pi^4/45 - 4*Pi^6/945 + 10*zeta(3) + 6*zeta(5) + 2*zeta(7).
Sum_{n>=1} (-1)^(n+1)/a(n) = 28*log(2) + 15*zeta(3)/2 + 45*zeta(5)/8 + 63*zeta(7)/32 - 16 - 5*Pi^2/6 - 7*Pi^4/90 - 31*Pi^6/7560. (End)

Extended by R. J. Mathar, Feb 05 2010

A082289 Expansion of x^4(2+x)/((1+x)(1-x)^5).

Original entry on oeis.org

2, 9, 26, 59, 116, 206, 340, 530, 790, 1135, 1582, 2149, 2856, 3724, 4776, 6036, 7530, 9285, 11330, 13695, 16412, 19514, 23036, 27014, 31486, 36491, 42070, 48265, 55120, 62680, 70992, 80104, 90066, 100929, 112746, 125571, 139460, 154470

Offset: 4

Michael Somos, Apr 07 2003

Vincenzo Librandi, Table of n, a(n) for n = 4..10000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A070893, A082290.

Cf. A045947 (which contains the first differences). - Bruno Berselli, Aug 26 2011

[(1/96)*(2*(n-2)*n*(3*n^2-10*n+4)+3*(-1)^n-3): n in [4..50]]; // Vincenzo Librandi, Aug 29 2011

Drop[CoefficientList[Series[x^4(2+x)/((1+x)(1-x)^5),{x,0,50}],x],4] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{2,9,26,59,116,206},50] (* Harvey P. Dale, Aug 26 2013 *)

a(n)=polcoeff(if(n>0,x^4*(2+x)/((1+x)*(1-x)^5),x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)),abs(n))

G.f.: x^4*(2+x)/((1+x)*(1-x)^5).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 3 by this equation, then a(n)=0 for 0 <= n <= 3 and a(n) = A070893(-n) for n < 0.
a(n) = A082290(2*n-7).
a(n) = (1/96)*(2*(n-2)*n*(3*n^2 - 10*n + 4) + 3*(-1)^n - 3). a(n) - a(n-2) = A006002(n-3) for n > 5. - Bruno Berselli, Aug 26 2011
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6); a(4)=2, a(5)=9, a(6)=26, a(7)=59, a(8)=116, a(9)=206. - Harvey P. Dale, Aug 26 2013

cp's OEIS Frontend

A163274 a(n) = n^4*(n+1)^2/2.

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A163275 a(n) = n^5*(n+1)^2/2.

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A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

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A267370 Partial sums of A140091.

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A317303 Numbers k such that both Dyck paths of the symmetric representation of sigma(k) have a central peak.

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A330298 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 2 even numbers.

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A082146 Expansion of g.f.: (1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).

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A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

A082146 Expansion of g.f.: (1+x^5)/((1-x^2)(1-x^3)(1-x^4)*(1-x^6)).

A082289 Expansion of x^4(2+x)/((1+x)(1-x)^5).