A027656 -id:A027656 - cp's OEIS frontend

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

1, 3, 9, 20, 42, 78, 139, 231, 372, 573, 861, 1254, 1791, 2499, 3432, 4629, 6162, 8085, 10492, 13455, 17094, 21503, 26832, 33201, 40795, 49764, 60333, 72687, 87096, 103785, 123075, 145236, 170646, 199626, 232617, 269997, 312277, 359898, 413448, 473438

Offset: 1

Alford Arnold, Aug 21 2009

Convolution of A006918 with A001399, or of A002625 with A059841 (A000035 if offsets are respected),
or of A038163 with A022003 or of A057524 with A027656 or of A014125 with the aerated version of A000217,
or of A002624 with A103221, or of A002623 with A008731, or of other combinations of splitting the signature -/3,3,1 into two components.
If we apply the enumeration of Molien series as described in A139672,
this is row 45=9*5 of a table of values related to Molien series, i.e., the
product of the sequence on row 9 (A006918) with the sequence on row 5 (A001399).
This is associated with the root system E6, and can be described using the additive function on the affine E6 diagram:
      1
      |
      2
      |
1--2--3--2--1

			To calculate a(3), we consider the first three terms of A001399 = (1 1 2...)
and the first three terms of A006918 = (1 2 5 ...), to get the convolved a(3) = 1*5+1*2+2*1 = 9.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Cf. A139672 (row 21).
For G2, the corresponding sequence is A001399.
For F4, the corresponding sequence is A115264.
For E7, the corresponding sequence is A210068.
For E8, the corresponding sequence is A045513.
See A210634 for a closely related sequence.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^3*(1-x^2)^3*(1-x^3)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(x/((1-x)^3*(1-x^2)^3*(1-x^3)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Jan 13 2020

Rest@CoefficientList[Series[x/((1-x)^3*(1-x^2)^3*(1-x^3)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{1,3,9,20,42,78,139,231,372,573,861,1254},40] (* Harvey P. Dale, Aug 03 2025 *)

Vec(1/(1-x)^3/(1-x^2)^3/(1-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

x=PowerSeriesRing(QQ, 'x', 40).gen()
1/((1-x)^3*(1-x^2)^3*(1-x^3))

a(n) = round( -(-1)^n*(n+3)*(n+7)/256 +(6*n^6 +180*n^5 +2070*n^4 +11400*n^3 +30429*n^2 +34290*n +9785)/103680 ) - R. J. Mathar, Mar 19 2012

Edited and extended by R. J. Mathar, Aug 22 2009

Corrected link to index entries - R. J. Mathar, Aug 26 2009

A003451 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 4, 8, 16, 25, 40, 56, 80, 105, 140, 176, 224, 273, 336, 400, 480, 561, 660, 760, 880, 1001, 1144, 1288, 1456, 1625, 1820, 2016, 2240, 2465, 2720, 2976, 3264, 3553, 3876, 4200, 4560, 4921, 5320, 5720, 6160, 6601, 7084, 7568, 8096, 8625, 9200, 9776, 10400

Offset: 5

N. J. A. Sloane

nonn

In other words, the number of 2-dissections of an n-gon modulo the cyclic action.

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

D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Column 3 of A295633.

Cf. A003453, A006584, A027656.

[(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // Vincenzo Librandi, Apr 05 2015

T51:= proc(n)
if n mod 2 = 0 then n*(n-2)*(n-4)/12;
else (n+1)*(n-3)*(n-4)/12; fi end;
[seq(T51(n),n=5..80)]; # N. J. A. Sloane, Dec 28 2012

Table[((n - 4) (2 n^2 - 4 n - 3 (1 - (-1)^n)) / 24), {n, 5, 60}] (* Vincenzo Librandi, Apr 05 2015 *)
CoefficientList[Series[(1+2*x-x^2)/((1-x)^4*(1+x)^2),{x,0,20}],x] (* Vaclav Kotesovec, Apr 05 2015 *)

Vec((1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Michel Marcus, Apr 04 2015

\\ See A295495 for DissectionsModCyclic()
{ my(v=DissectionsModCyclic(apply(i->y, [1..30]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

G.f.: x^5 * (1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2).
See also the Maple code for an explicit formula.
a(n) = A006584(n+3) - A027656(n). - Yosu Yurramendi, Aug 07 2008
a(n) = (n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24, for n>=5. - Luce ETIENNE, Apr 04 2015

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012
First formula adapted to offset by Vaclav Kotesovec, Apr 05 2015
Name clarified by Andrew Howroyd, Nov 25 2017

A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 2, 2, 1, 3, 2, 3, 3, 1, 0, 3, 4, 5, 4, 1, 4, 3, 5, 7, 8, 5, 1, 0, 4, 6, 9, 12, 12, 6, 1, 5, 4, 7, 11, 16, 20, 17, 7, 1, 0, 5, 8, 13, 20, 28, 32, 23, 8, 1, 6, 5, 9, 15, 24, 36, 48, 49, 30, 9, 1, 0, 6, 10, 17, 28, 44, 64, 80, 72, 38, 10, 1, 7, 6, 11, 19, 32, 52, 80, 112, 129

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n,0)=T(n,2) by T(2n,0)=T(n,m) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058393, A058395, A057884 (and effectively A007318).

			Rows are (1,0,2,0,3,0,4,...), (1,1,2,2,3,3,...), (1,2,3,4,5,6,...), (1,3,5,7,9,11,...), etc.

Rows are A027656 (A000027 with zeros), A008619, A000027, A005408, A008574 etc. Columns are A000012, A001477, A022856 etc. Diagonals include A034007, A045891, A045623, A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 etc. The triangle A055249 also appears in half of the array.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(2n, 0)=T(n, 2) and T(2n+1, 0)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^2.

A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 3, 0, 2, 0, 1, 0, 3, 0, 2, 0, 1, 4, 0, 3, 0, 2, 0, 1, 0, 4, 0, 3, 0, 2, 0, 1, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1

Offset: 1

Gary W. Adamson, Mar 31 2009

Eigensequence of the triangle = A158943: (1, 1, 3, 5, 10, 19, 36, 69, 131, ...)

			First few rows of the triangle =
  1;
  0, 1;
  2, 0, 1;
  0, 2, 0, 1;
  3, 0, 2, 0, 1;
  0, 3, 0, 2, 0, 1;
  4, 0, 3, 0, 2, 0, 1;
  0, 4, 0, 3, 0, 2, 0, 1;
  5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  ...
The inverse array begins
   1;
   0,  1;
  -2,  0,  1;
   0, -2,  0,  1;
   1,  0, -2,  0,  1;
   0,  1,  0, -2,  0,  1;
   0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  0,  1,  0, -2,  0,  1;
   ... - _Peter Bala_, Aug 15 2021

D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Cf. A158943, A158945, A156663.

seq(seq((1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # Peter Bala, Aug 15 2021

Triangle by columns: A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) in every column.
From Peter Bala, Aug 15 2021: (Start)
T(n,k) = (1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
Double Riordan array (1/(1-x)^2; x, x) as defined in Davenport et al.
The m-th power of the array is the double Riordan array (1/(1 - x)^(2*m); x, x). Cf. A156663. (End)

A166445 Hankel transform of A025276.

Original entry on oeis.org

1, 0, -1, 1, 3, 0, -3, 1, 5, 0, -5, 1, 7, 0, -7, 1, 9, 0, -9, 1, 11, 0, -11, 1, 13, 0, -13, 1, 15, 0, -15, 1, 17, 0, -17, 1, 19, 0, -19, 1, 21, 0, -21, 1, 23, 0, -23, 1, 25, 0, -25, 1, 27, 0, -27, 1, 29, 0, -29, 1, 31, 0, -31, 1, 33, 0, -33, 1, 35, 0, -35, 1

Offset: 0

Paul Barry, Oct 13 2009

Felix Fröhlich, Table of n, a(n) for n = 0..10000
Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,-2,2,-1,1).

Cf. A025276, A027656, A056594, A127365.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (1-x+x^2+x^4)/((1-x)*(1+x^2)^2) )); // G. C. Greubel, Jul 27 2024

LinearRecurrence[{1,-2,2,-1,1},{1,0,-1,1,3},80] (* Harvey P. Dale, Nov 21 2014 *)

Vec((1-x+x^2+x^4)/((1-x)*(1+x^2)^2) + O(x^80)) \\ Felix Fröhlich, Jun 28 2022

def A166445_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2+x^4)/((1-x)*(1+x^2)^2) ).list()
A166445_list(80) # G. C. Greubel, Jul 27 2024

G.f.: (1-x+x^2+x^4)/((1-x)*(1+x^2)^2).
a(n) = (1/2)*(1 + cos((n+1)*Pi/2) + (n+1)*sin((n+1)*Pi/2)). - Harvey P. Dale, Nov 21 2014; corrected by Bernard Schott, Jun 27 2022
For n >= 0: a(4n) = 2n+1; a(4n+1) = 0; a(4n+2) = -a(4n) = -2n-1; a(4n+3) = 1. - Bernard Schott, Jun 27 2022
a(n) - a(n-1) = A127365(n+1). - R. J. Mathar, Jul 01 2024
E.g.f.: (exp(x) + cos(x) - (1 + x)*sin(x))/2. - Stefano Spezia, Jul 14 2024
a(n) = (1/2)*(1 - A056594(n) - A056594(n-1) + 2*(-1)^floor(n/2) * A027656(n)). - G. C. Greubel, Jul 27 2024

A186111 a(n) = -n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

Original entry on oeis.org

1, -3, 3, -2, 5, -9, 7, -4, 9, -15, 11, -6, 13, -21, 15, -8, 17, -27, 19, -10, 21, -33, 23, -12, 25, -39, 27, -14, 29, -45, 31, -16, 33, -51, 35, -18, 37, -57, 39, -20, 41, -63, 43, -22, 45, -69, 47, -24, 49, -75, 51, -26, 53, -81, 55, -28, 57, -87, 59, -30, 61, -93, 63

Offset: 1

Michael Somos, Feb 13 2011

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (-2,-3,-4,-3,-2,-1).

Cf. A068073, A186690, A186813.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)^3*(1-x^3)/(1-x^4)^2)); // G. C. Greubel, Aug 14 2018

Rest[CoefficientList[Series[x (1-x)^3(1-x^3)/(1-x^4)^2,{x,0,70}],x]] (* or *) LinearRecurrence[{-2,-3,-4,-3,-2,-1},{1,-3,3,-2,5,-9},70] (* Harvey P. Dale, Aug 08 2012 *)
a[ n_] := n If[ OddQ[n], 1, -(Mod[n/2, 2] + 1/2)]; (* Michael Somos, Apr 25 2015 *)
a[ n_] := n {1, -3/2, 1, -1/2}[[Mod[n, 4, 1]]]; (* Michael Somos, Apr 25 2015 *)

{a(n) = -(-1)^n * n * [1, 2, 3, 2] [n%4 + 1] / 2};

{a(n) = sign(n) * polcoeff( x * (1 - x)^3 * (1 - x^3) / (1 - x^4)^2 + x * O(x^abs(n)), abs(n))};

{a(n) = n * if( n%2, 1, -(n/2%2 + 1/2))}; /* Michael Somos, Apr 25 2015 */

a(n) is multiplicative with a(2) = -3, a(2^e) = -(2^(e-1)) if e>1, a(p^e) = p^e if p>2.
Euler transform of length 4 sequence [-3, 0, -1, 2].
G.f.: x * (1 - x)^3 * (1 - x^3) / (1 - x^4)^2.
G.f.: x * (1 + x + x^2) * (1 - x)^2 / ((1 + x)^2 * (1 + x^2)^2).
Dirichlet g.f. zeta(s-1)*( 1-5*2^(-s)+4^(1-s)). - R. J. Mathar, Mar 31 2011
a(n) = (-1)^(n+1)*n + (-1)^floor(n/2)*A027656(n-2). - R. J. Mathar, Mar 31 2011
a(n) = -2*a(n-1) - 3*a(n-2) - 4*a(n-3) - 3*a(n-4) - 2*a(n-5) - a(n-6) with a(1)=1, a(2)=-3, a(3)=3, a(4)=-2, a(5)=5, a(6)=-9. - Harvey P. Dale, Aug 08 2012
G.f.: 1/(1+x) - 1/(1+x)^2 - 1/(1+x^2) + 1/(1+x^2)^2. - Michael Somos, Apr 24 2015
a(n) = -a(-n) for all n in Z. - Michael Somos, Apr 24 2015
G.f.: f(x) - f(x^2) where f(x) := x / (1 + x)^2. - Michael Somos, May 07 2015
Moebius transform of A186690. - Michael Somos, Apr 25 2015
a(n) = -(-1)^n * A186813(n). - Michael Somos, May 07 2015
a(n) = n*cos(n*Pi/2)/2-n*(-1)^n. - Wesley Ivan Hurt, May 05 2021

A058936 Decomposition of Stirling's S(n,2) based on associated numeric partitions.

Original entry on oeis.org

0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260, 45360, 25920, 20160, 18144, 403200, 226800, 172800, 151200, 72576, 3991680, 2217600, 1663200, 1425600, 1330560, 43545600, 23950080, 17740800, 14968800, 13685760, 6652800, 518918400

Offset: 1

Alford Arnold, Jan 11 2001

These values also appear in a wider context when counting elements of finite groups by cycle structure. For example, the alternating group on four symbols has 12 elements; eight associated with the partition 3+1, three associated with 2+2 and the identity associated with 1+1+1+1. The cross-referenced sequences are all associated with similar numeric partitions and "M2" weights.

			Triangle begins:
  0;
  1;
  3;
  8, 3;
  30, 20;
  144, 90, 40;
  840, 504, 420;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000012, A000035, A000027, A004526, A022003, A008619, A000217, A007997, A001399, A011765 A008620, A027656, A002620, A000292, A008627.

From Sean A. Irvine, Sep 05 2022: (Start)
T(1,1) = 0.
T(n,k) = n! / (k * (n-k)) for 1 <= k < n/2.
T(2n,n) = (2*n)! / (2*n^2).
(End)

More terms from Sean A. Irvine, Sep 05 2022

A062534 Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).

Original entry on oeis.org

1, -2, 1, 3, -1, 1, -4, 2, 0, 1, 5, -2, 2, 1, 1, -6, 3, 0, 3, 2, 1, 7, -3, 3, 3, 5, 3, 1, -8, 4, 0, 6, 8, 8, 4, 1, 9, -4, 4, 6, 14, 16, 12, 5, 1, -10, 5, 0, 10, 20, 30, 28, 17, 6, 1, 11, -5, 5, 10, 30, 50, 58, 45, 23, 7, 1, -12, 6, 0, 15, 40, 80, 108, 103, 68, 30, 8, 1, 13, -6, 6, 15, 55, 120, 188, 211, 171, 98, 38, 9, 1, -14, 7, 0, 21, 70, 175

Offset: 0

Henry Bottomley, Jun 25 2001

Rows are effectively (with minor adjustments): A038608, A001057, A027656, A008805, A006918, A002624, A028346. Cf. A058394 which (adjusting for signs and an overlap of three rows) is effectively the continuation of this table for negative n.

Each row is partial sum of preceding row, i.e. T(n, k)=T(n-1, k)+T(n, k-1) with T(0, k)=(k+1)*(-1)^k and T(n, 0)=1.

A092093 Back and Forth Summant S(n, 5): a(n) = sum{i = 0..floor(2n/5)} n-5i.

Original entry on oeis.org

1, 2, 1, 3, 0, 3, 6, 2, 6, 0, 5, 10, 3, 9, 0, 7, 14, 4, 12, 0, 9, 18, 5, 15, 0, 11, 22, 6, 18, 0, 13, 26, 7, 21, 0, 15, 30, 8, 24, 0, 17, 34, 9, 27, 0, 19, 38, 10, 30, 0, 21, 42, 11, 33, 0, 23, 46, 12, 36, 0, 25, 50, 13, 39, 0, 27, 54, 14, 42, 0, 29, 58, 15, 45, 0, 31, 62, 16, 48, 0, 33

Offset: 1

Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004

J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004.
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.

J. Dezert, Smaran dacheials
J. Dezert, S marandacheials, "Mathematics Magazine", Canada
F. Smarandache, Summants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A092096, A092399.

Other values of k: A000004 (k = 1, 2), A092092 (k = 3), A027656 (k = 4).

S(n, k=5) = local(s, x); s = n; x = n - k; while (x >= -n, s = s + x; x = x - k); s;

a(5n) = 0; a(5n+1) = 2n+1; a(5n+2) = 4n+2; a(5n+3) = n+1; a(5n+4) = 3n+3.

G.f.: x*(2*x^6+x^5+3*x^3+x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, Jul 28 2013

Edited and extended by David Wasserman, Dec 19 2005

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 4, 9, 21, 35, 65, 95, 155, 210, 315, 406, 574, 714, 966, 1170, 1530, 1815, 2310, 2695, 3355, 3861, 4719, 5369, 6461, 7280, 8645, 9660, 11340, 12580, 14620, 16116, 18564, 20349, 23256, 25365, 28785, 31255, 35245, 38115, 42735, 46046, 51359, 55154, 61226, 65550, 72450, 77350, 85150, 90675, 99450, 105651, 115479

Offset: 0

Ilya Gutkovskiy, Aug 15 2017

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

G.f.: x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A085787.
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - Luce ETIENNE, Nov 18 2017

cp's OEIS Frontend

A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).

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A003451 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.

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A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

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A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)

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A166445 Hankel transform of A025276.

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A186111 a(n) = -n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

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A058936 Decomposition of Stirling's S(n,2) based on associated numeric partitions.

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A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).