A001834 -id:A001834 - cp's OEIS frontend

A324176 Integers k such that floor(sqrt(k)) + floor(sqrt(k/3)) divides k.

1, 2, 6, 15, 18, 24, 32, 36, 45, 55, 72, 78, 84, 98, 105, 112, 136, 144, 152, 180, 198, 220, 230, 275, 336, 390, 403, 462, 525, 540, 608, 663, 697, 756, 774, 792, 836, 855, 874, 940, 980, 1050, 1092, 1144, 1166, 1265, 1368, 1392, 1500, 1525, 1586, 1638, 1755, 1782, 1848, 1904

Offset: 1

Jinyuan Wang, Mar 08 2019

nonn

This sequence is infinite for the same reason that A324175 is: if x-1 > y > 1 satisfies x^2 - 3*y^2 = -2 (x=A001834(j), y=A001835(j+1), j>0), then x < 3*y. Let k = 3*y^2 + m. By the pigeonhole principle there exists a number m belonging to [0, 2*x - 2] such that x + y | 3*y^2 + m, so such a k is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001834, A001835, A324175.

Select[Range[2000],Divisible[#,Floor[Sqrt[#]]+Floor[Sqrt[#/3]]]&] (* Harvey P. Dale, Jun 19 2021 *)

is(n) = n%(floor(sqrt(n)) + floor(sqrt(n/3))) == 0;

A001352 a(0) = 1, a(1) = 6, a(2) = 24; for n>=3, a(n) = 4a(n-1) - a(n-2).

Original entry on oeis.org

1, 6, 24, 90, 336, 1254, 4680, 17466, 65184, 243270, 907896, 3388314, 12645360, 47193126, 176127144, 657315450, 2453134656, 9155223174, 34167758040, 127515808986, 475895477904, 1776066102630, 6628368932616, 24737409627834, 92321269578720, 344547668687046

Offset: 0

N. J. A. Sloane

Also the coordination sequence of a {4,6} tiling of the hyperbolic plane, where there are 6 squares (with vertex angles Pi/3) around every vertex. - toen (tca110(AT)rsphysse.anu.edu.au), May 16 2005

a(n) is related to the almost-equilateral Heronian triangles because it is the area of the Heronian triangle with edge lengths A003500(n)-1, A003500(n)+1 and 4. - Herbert Kociemba, Mar 19 2021

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012
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).

T. D. Noe, Table of n, a(n) for n = 0..200
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
D. Fortin, B-spline Toeplitz inverse under corner perturbations, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From _N. J. A. Sloane_, Oct 22 2012
T. N. E. Greville, Table for third-degree spline interpolations with equally spaced arguments, Math. Comp., 24 (1970), 179-183.
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 (4,-1).

First differences of A082841. Pairwise sums of A001834.

A001352 := proc(n) coeftayl(1+6*x/(1-4*x+x^2),x=0,n) ; end: for n from 0 to 30 do printf("%d,",A001352(n)) ; od ; # R. J. Mathar, Jun 06 2007
A001352:=(z+1)**2/(1-4*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

Join[{1},LinearRecurrence[{4,-1},{6,24},30]] (* Harvey P. Dale, Jul 20 2011 *)

Vec((x+1)^2/(x^2-4*x+1) + O(x^40)) \\ Colin Barker, Oct 12 2015

G.f.: 1+6x/(1-4x+x^2). - R. J. Mathar, Jun 06 2007

a(n) = sqrt(3)*(-(2-sqrt(3))^n+(2+sqrt(3))^n) for n>0. - Colin Barker, Oct 12 2015

More terms from R. J. Mathar, Jun 06 2007

A121338 Pentagonal numbers P(k) that are one-third of other pentagonal numbers: P(k) such that 3*P(k)=P(m) for some m>k.

Original entry on oeis.org

70, 511258935, 3732600255368600, 27250975409595074561065, 198953975772318806945317308330, 1452523584226469439408576900215922395, 10604587088767577582197244731443261336155260, 77421990626847055423676582260371016672624778798925

Offset: 1

Franz Vrabec, Aug 28 2006

The k values are (A001835(6n-2)+1)/6, the m values are (A001834(6n-3)+1)/6.

			a(1) = ((A001835(4))^2-1)/24 = (41^2-1)/24 = 70; this number and 3*70=210 are pentagonal numbers (in A000326).

Colin Barker, Table of n, a(n) for n = 1..146
Index entries for linear recurrences with constant coefficients, signature (7300803,-7300803,1).

Cf. A001834, A001835.

CoefficientList[Series[5 (x^2 + 40545 x + 14)/((1 - x) (x^2 - 7300802 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 21 2015 *)

Vec(-5*x*(x^2+40545*x+14)/((x-1)*(x^2-7300802*x+1)) + O(x^20)) \\ Colin Barker, Jun 20 2015

a(n) = ((A001835(6n-2))^2-1)/24.
a(n) = 7300803*a(n-1)-7300803*a(n-2)+a(n-3). - Colin Barker, Jun 20 2015
G.f.: -5*x*(x^2+40545*x+14) / ((x-1)*(x^2-7300802*x+1)). - Colin Barker, Jun 20 2015

Added more terms, Colin Barker, Jun 20 2015

A207607 Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 9, 9, 2, 1, 14, 25, 13, 2, 1, 20, 55, 49, 17, 2, 1, 27, 105, 140, 81, 21, 2, 1, 35, 182, 336, 285, 121, 25, 2, 1, 44, 294, 714, 825, 506, 169, 29, 2, 1, 54, 450, 1386, 2079, 1716, 819, 225, 33, 2, 1, 65, 660, 2508, 4719, 5005, 3185, 1240, 289, 37, 2

Offset: 1

Clark Kimberling, Feb 19 2012

Subtriangle of the triangle T(n,k) given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First five rows:
  1;
  1,  2;
  1,  5,  2;
  1,  9,  9,  2;
  1, 14, 25, 13,  2;
Triangle (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  5,  2,  0;
  1,  9,  9,  2,  0;
  1, 14, 25, 13,  2,  0;
  1, 20, 55, 49, 17,  2,  0;
  ...
1 = 2*1 - 1, 20 = 2*14 + 1 - 9, 55 = 2*25 + 14 - 9, 49 = 2*13 + 25 - 2, 17 = 2*2 + 1 - 0, 2 = 2*0 + 2 - 0. - _Philippe Deléham_, Mar 03 2012

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A207606.

A207607:= (n,k) -> `if`(k=1, 1, binomial(n+k-3, 2*k-2) + 2*binomial(n+k-3, 2*k-3) ); seq(seq(A207607(n, k), k = 1..n), n = 1..10); # G. C. Greubel, Mar 15 2020

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207606 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207607 *)
(* Second program *)
Table[If[k==1, 1, Binomial[n+k-3, 2*k-2] + 2*Binomial[n+k-3, 2*k-3]], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Mar 15 2020 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

def T(n, k):
    if k == 1: return 1
    else: return binomial(n+k-3, 2*k-2) + 2*binomial(n+k-3, 2*k-3)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 15 2020

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = x*u(n-1,x) + (x+1)v(n-1,x), where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Mar 03 2012
G.f.: (1-x+y*x)/(1-(y+2)*x+x^2). - Philippe Deléham, Mar 03 2012
For n >= 1, Sum{k=0..n} T(n,k)*x^k = A000012(n), A001906(n), A001834(n-1), A055271(n-1), A038761(n-1), A056914(n-1) for x = 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Mar 03 2012
T(n,k) = C(n+k-1,2*k) + 2*C(n+k-1,2*k-1). where C is binomial. - Yuchun Ji, May 23 2019
T(n,k) = T(n-1,k) + A207606(n,k-1). - Yuchun Ji, May 28 2019
Sum_{k=1..n} T(n, k)*x^k = { 4*(-1)^(n-1)*A016921(n-1) (x=-4), 3*(-1)^(n-1) * A130815(n-1) (x=-3), 2*(-1)^(n-1)*A010684(n-1) (x=-2), A057079(n+1) (x=-1), 0 (x=0), A001906(n) = Fibonacci(2*n) (x=1), 2*A001834(n-1) (x=2), 3*A055271(n-1) (x=3), 4*A038761(n-1) (x=4) }. - G. C. Greubel, Mar 15 2020

A209759 Triangle of coefficients of polynomials u(n,x) jointly generated with A209760; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 5, 16, 13, 1, 5, 19, 48, 34, 1, 5, 19, 68, 141, 89, 1, 5, 19, 71, 233, 409, 233, 1, 5, 19, 71, 262, 772, 1175, 610, 1, 5, 19, 71, 265, 948, 2492, 3349, 1597, 1, 5, 19, 71, 265, 986, 3354, 7879, 9482, 4181, 1, 5, 19, 71, 265, 989, 3641

Offset: 1

Clark Kimberling, Mar 14 2012

Limiting row: A001834
Coefficient of x^n in u(n,x):  odd-indexed Fibonacci numbers
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...; A033999
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...5...5
1...5...16...13
1...5...19...48...34
First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.

Cf. A209760, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209760 *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210224 Triangle of coefficients of polynomials v(n,x) jointly generated with A210223; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 5, 15, 7, 1, 5, 19, 35, 9, 1, 5, 19, 63, 69, 11, 1, 5, 19, 71, 177, 121, 13, 1, 5, 19, 71, 249, 429, 195, 15, 1, 5, 19, 71, 265, 781, 923, 295, 17, 1, 5, 19, 71, 265, 957, 2171, 1807, 425, 19, 1, 5, 19, 71, 265, 989, 3211, 5407, 3281, 589, 21

Offset: 1

Clark Kimberling, Mar 20 2012

Limiting row: A001834

For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...3
1...5...5
1...5...15...7
1...5...19...35...9
First three polynomials v(n,x): 1, 1 + 3x , 1 + 5x + 5x^2.

Cf. A210223, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]     (* A210223 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210224 *)

u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A386706 Expansion of ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.

Original entry on oeis.org

0, 1, 5, 18, 71, 260, 990, 3672, 13775, 51343, 191860, 715770, 2672298, 9972092, 37220040, 138903480, 518408351, 1934712530, 7220497115, 26947209762, 100568547820, 375326739216, 1400739172470, 5227629044040, 19509779871450, 72811487038701, 271736178975820, 1014133216234068

Offset: 0

Christian Kassel, Jul 30 2025

nonn

a(n) is the value at q = 2 + sqrt(3) of C_n(q)/(q^{n-1}(q - 1)^2), where C_n(q) is the number of codimension n ideals of the algebra of two-variable Laurent polynomials over a finite field of order q. The number C_n(q) is a palindromic polynomial of degree 2n with integer coefficients in the variable q and it is divisible by (q-1)^2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Christian Kassel and Christophe Reutenauer, Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables, arXiv:1505.07229 [math.AG], 2015-2016; Michigan Math. J. 67 (2018), 715-741.
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016; The Ramanujan Journal 46 (2018), 633-655.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025.

Cf. A001834, A329156, A387017.

nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^2/(1 - 4*x^k + x^(2*k)), {k, 1, nmax}] - 1)/2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 30 2025 *)

G.f.: ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.
a(2^k) = A001834(2^k-1) for all nonnegative integers k. Follows from Cor. 4.5 of Kassel-Reutenauer paper "Pairs of intertwined integer sequences".
a(n) ~ (1 + sqrt(3))^(2*n-1) / 2^n. - Vaclav Kotesovec, Jul 30 2025

a(0)=0 added, offset changed to 0, a(7) corrected and more terms added by Vaclav Kotesovec, Jul 30 2025

A109438 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) + 2*(-1)^(n+1), alternatively a(n) = 3a(n-1) + 3a(n-2) - a(n-3).

Original entry on oeis.org

1, 5, 18, 68, 253, 945, 3526, 13160, 49113, 183293, 684058, 2552940, 9527701, 35557865, 132703758, 495257168, 1848324913, 6898042485, 25743845026, 96077337620, 358565505453, 1338184684193, 4994173231318, 18638508241080

Offset: 0

Creighton Dement, Jun 28 2005

See A109437 for comments.

Floretion Algebra Multiplication Program, FAMP Code: (-1)^(n)jbasejfor[ + .5'ii' + .5'kk' + .5'ij' + .5'ji' + .5'jk' + .5'kj'] 1vesfor = (-1,-1,-1,-1,)

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

Cf. A109437, A001834, A102871.

LinearRecurrence[{3,3,-1},{1,5,18},30] (* Harvey P. Dale, Sep 07 2021 *)

Vec((1 + 2*x) / ((1 + x)*(1 - 4*x + x^2)) + O(x^30)) \\ Colin Barker, May 12 2019

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

a(n) = (-2*(-1)^n + (7-5*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(7+5*sqrt(3))) / 12. - Colin Barker, May 12 2019

A129743 a(n) = -(u^n-1)*(v^n-1) with u = 2+sqrt(3), v = 2-sqrt(3).

Original entry on oeis.org

2, 12, 50, 192, 722, 2700, 10082, 37632, 140450, 524172, 1956242, 7300800, 27246962, 101687052, 379501250, 1416317952, 5285770562, 19726764300, 73621286642, 274758382272, 1025412242450, 3826890587532, 14282150107682, 53301709843200, 198924689265122, 742397047217292

Offset: 1

N. J. A. Sloane, May 13 2007

Each term of this sequence beyond the sixth has a primitive prime divisor. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007

a(n) is also the number of spanning trees for the n-gear graph. - Eric W. Weisstein, Jul 16 2011

Stefano Spezia, Table of n, a(n) for n = 1..1700
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
Anthony Flatters, Primitive Divisors of some Lehmer-Pierce Sequences, arXiv:0708.2190 [math.NT], 2007.
Eric Weisstein's World of Mathematics, Gear Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001353, A001834, A092184.

u:=2+sqrt(3): v:=2-sqrt(3): a:=n->expand(-(u^n-1)*(v^n-1)): seq(a(n),n=1..28); # Emeric Deutsch, May 13 2007

Table[-((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1), {n, 30}] // Expand (* Stefan Steinerberger, May 15 2007 *)
LinearRecurrence[{5, -5, 1}, {2, 12, 50}, 30]
LucasL[2 Range[20], Sqrt[2]] - 2 // Round (* Eric W. Weisstein, Mar 28 2018 *)

my(x='x+O('x^30)); Vec(2*x*(1+x)/((1-x)*(1-4*x+x^2))) \\ Altug Alkan, Mar 28 2018

a(2*n) = 12*A001353(n)^2, a(2*n+1) = 2*A001834(n)^2. - Vladeta Jovovic, May 30 2007
a(n) = 2*A092184(n). - Robert G. Wilson v, Jul 04 2007
O.g.f.: 2*x*(1+x)/((1-x)*(1-4*x+x^2)). - R. J. Mathar, Dec 05 2007
a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). - Eric W. Weisstein, Jul 15 2011
E.g.f.: 2*exp(x)*(exp(x)*cosh(sqrt(3)*x) - 1). - Stefano Spezia, May 05 2024

More terms from Emeric Deutsch and Stefan Steinerberger, May 13 2007

More terms from Vladeta Jovovic, May 30 2007

A133161 Indices of the triangular numbers which are also centered triangular number.

Original entry on oeis.org

1, 4, 16, 61, 229, 856, 3196, 11929, 44521, 166156, 620104, 2314261, 8636941, 32233504, 120297076, 448954801, 1675522129, 6253133716, 23337012736, 87094917229, 325042656181, 1213075707496, 4527260173804, 16895964987721

Offset: 1

Richard Choulet, Oct 09 2007

Also, indices of the triangular numbers which are sums of three consecutive triangular numbers (see A129803).

A. Kozikowska, Topological classes of statically determinate beams with arbitrary number of supports, JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 4, pp. 1079-1100, Warsaw 2011; (see Eq. 5.18). - _N. J. A. Sloane_, Dec 17 2011.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001834, A102871, A128862, A129803, A001571.

LinearRecurrence[{5,-5,1},{1,4,16},30] (* Harvey P. Dale, Aug 29 2017 *)

a(n+2)=4*a(n+1)-a(n)+1.
a(n+1)=2*a(n)+0.5+0.5*(12*a(n)^2+12*a(n)-15)^0.5.
G.f.: x*(1-x+x^2)/(1-x)/(1-4*x+x^2). - R. J. Mathar, Oct 24 2007
a(n)-a(n-1)= A005320(n-1). - R. J. Mathar, Mar 14 2016

cp's OEIS Frontend

A324176 Integers k such that floor(sqrt(k)) + floor(sqrt(k/3)) divides k.

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A001352 a(0) = 1, a(1) = 6, a(2) = 24; for n>=3, a(n) = 4a(n-1) - a(n-2).

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A121338 Pentagonal numbers P(k) that are one-third of other pentagonal numbers: P(k) such that 3*P(k)=P(m) for some m>k.

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A207607 Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.

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A209759 Triangle of coefficients of polynomials u(n,x) jointly generated with A209760; see the Formula section.

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A210224 Triangle of coefficients of polynomials v(n,x) jointly generated with A210223; see the Formula section.

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A386706 Expansion of ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.

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