A108368 -id:A108368 - cp's OEIS frontend

A108369 Coefficients of x/(1+3x+3x^2-x^3).

0, 1, -3, 6, -8, 3, 21, -80, 180, -279, 217, 366, -2028, 5203, -9159, 9840, 3160, -48159, 144837, -286874, 377952, -128397, -1035539, 3869760, -8631060, 13248361, -9982143, -18429714, 98483932, -250144797, 436552881, -460740320, -177582480, 2351521281

Offset: 0

Michael Somos, Jun 01 2005

sign

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 562.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Vincent Thill, Radicaux et Ramanujan, April 2021, see c(n).
Index entries for linear recurrences with constant coefficients, signature (-3,-3,1).

Cf. A000748, A108368.

CoefficientList[Series[x/(1+3x+3x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-3,-3,1},{0,1,-3},40] (* Harvey P. Dale, Jul 30 2024 *)

{a(n)=if(n>=0, polcoeff(x/(1+3*x+3*x^2-x^3)+x*O(x^n),n), n=-1-n; polcoeff(x/(1-3*x-3*x^2-x^3)+x*O(x^n),n))}

x=a(n), z=a(-n-2), y=a(n)+a(n+1), t=a(-1-n)+a(-n-2) is a solution to 2*(x^3+z^3) = y^3+t^3.
G.f.: x/(1+3*x+3*x^2-x^3).
a(n) = -3*a(n-1) - 3*a(n-2) + a(n-3).
a(-1-n) = A108368(n).
a(n+1) = (-1)^n * Sum_{k=0..floor(n/3)} (-2)^k * binomial(n+2,3*k+2). - Seiichi Manyama, Aug 05 2024

A377314 a(n) = coefficient of the term that is independent of 2^(1/3) and 2^(2/3) in the expansion of (1 + 2^(1/3) + 2^(2/3))^n.

Original entry on oeis.org

1, 1, 5, 19, 73, 281, 1081, 4159, 16001, 61561, 236845, 911219, 3505753, 13487761, 51891761, 199644319, 768096001, 2955112721, 11369270485, 43741245619, 168286661033, 647452990441, 2490960200041, 9583526232479, 36870912288001, 141854275761481

Offset: 0

Clark Kimberling, Oct 26 2024

nonn

See A377109 for a guide to related sequences.

			((1 + 2^(1/3) + 2^(2/3)))^3 = 19 + 15 2^(1/3) + 12 2^(2/3), so a(3) = 19.

Michael De Vlieger, Table of n, a(n) for n = 0..1709
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).

Cf. A377109, A377117, A377315, A108368 (coefficients of 2^(2/3)).

(* Program 1 generates sequences A377314-A377315 and A108368. *)
tbl = Table[Expand[(1 + 2^(1/3) + 2^(2/3))^n], {n, 0, 24}];
Take[tbl, 6]
u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
   Map[({#1, #1 /. ^ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
{s1, s2, s3} = Transpose[(PadRight[#1, 3] &) /@ Last /@ u][[1 ;; 3]];
s1  (* Peter J. C. Moses, Oct 16 2024 *)
(* Program 2 generates (a(n)) for n>=1. *)
LinearRecurrence[{3,3,1}, {1, 1, 5}, 15]

a(n) = 3*a(n-1) + 3*a(n-2) + a(n-3), with a(0)=1, a(1)=1, a(3)=5. [Corrected by Jianing Song, Oct 31 2024]

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

A377315 a(n) = coefficient of 2^(1/3) in the expansion of (1 + 2^(1/3) + 2^(2/3))^n.

Original entry on oeis.org

0, 1, 4, 15, 58, 223, 858, 3301, 12700, 48861, 187984, 723235, 2782518, 10705243, 41186518, 158457801, 609638200, 2345474521, 9023795964, 34717449655, 133569211378, 513883779063, 1977076420978, 7606449811501, 29264462476500, 112589813284981, 433169277095944

Offset: 0

Clark Kimberling, Oct 26 2024

See A377109 for a guide to related sequences.

			(1 + 2^(1/3) + 2^(2/3))^3 = 19 + 15 2^(1/3) + 12 2^(2/3), so a(3) = 15.

Index entries for linear recurrences with constant coefficients, signature (3,3,1).

Cf. A377109, A377117, A377314, A108368 (coefficients of 2^(2/3)).

(* Program 1 generates sequences A377314-A377315 and A108368. *)
tbl = Table[Expand[(1 + 2^(1/3) + 2^(2/3))^n], {n, 0, 24}];
Take[tbl, 6]
u = MapApply[{#1/#2, #2} /. {1, #} -> {{1}, {#}} &,
   Map[({#1, #1 /. ^ -> 1} &), Map[(Apply[List, #1] &), tbl]]];
{s1, s2, s3} = Transpose[(PadRight[#1, 3] &) /@ Last /@ u][[1 ;; 3]];
s2  (* Peter J. C. Moses, Oct 16 2024 *)
(* Program 2 generates (a(n)) for n>=1. *)
LinearRecurrence[{3,3,1}, {0, 1, 4}, 15]

a(n) = 3*a(n-1) + 3*a(n-2) + a(n-3), with a(0)=0, a(1)=1, a(3)=4.

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

A374454 Expansion of o.g.f. 1/(1 - 4x - 6x^2 - 4*x^3 - x^4).

Original entry on oeis.org

1, 4, 22, 116, 613, 3240, 17124, 90504, 478333, 2528092, 13361506, 70618412, 373233385, 1972618128, 10425707976, 55102092624, 291226324249, 1539193302772, 8134965235054, 42995028146468, 227237903531533, 1201000837247928, 6347545848001836, 33548135057767512

Offset: 0

Enrique Navarrete, Jul 08 2024

a(n) is the number of generalized compositions of n using parts of size at most 4 where there are binomial(4,i) types of i (see example).
The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i.
Related sequences that count the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i are A108368(n+1), A000129(n+1), and A000012(n) for k = 3, 2, 1, respectively.

			The following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 6 using parts of size at most 4 where there are binomial(4,i) types of i (ie. 4 types of 1, 6 types of 2, 4 types of 3 and 1 type of 4):
    Type                     Number              Total
    4+2                        2                    12
    3+3                        1                    16
    4+1+1                      3                    48
    3+2+1                      6                   576
    2+2+2                      1                   216
    3+1+1+1                    4                  1024
    2+2+1+1                    6                  3456
    2+1+1+1+1                  5                  7680
    1+1+1+1+1+1                1                  4096,
    adding to a(6) = 17124.

Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).

Cf. A000012, A000129, A108368, A374455.

Cf. A145840.

CoefficientList[Series[1/(1-4*x-6*x^2-4*x^3-x^4),{x,0,23}],x] (* Stefano Spezia, Jul 09 2024 *)

a(n) = 4*a(n-1) + 6*a(n-2) + 4*a(n-3) + a(n-4), n=>4.

a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(4*k,n). - Seiichi Manyama, Aug 03 2024

A374455 Expansion of o.g.f. 1/(1 - 5x - 10x^2 - 10x^3 - 5x^4 - x^5).

Original entry on oeis.org

1, 5, 35, 235, 1580, 10626, 71460, 480570, 3231845, 21734235, 146163251, 982951365, 6610371480, 44454906580, 298960311840, 2010515259876, 13520763292345, 90927457083265, 611489327404315, 4112280377388895, 27655184063541876, 185981775414350150, 1250731895575163300

Offset: 0

Enrique Navarrete, Jul 08 2024

a(n) is the number of generalized compositions of n using parts of size at most 5 where there are binomial(5,i) types of i (see example).

The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i.

			The following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 5 using parts of size at most 5 where there are binomial(5,i) types of i (ie. 5 types of 1, 10 types of 2, 10 types of 3, 5 types of 4, and 1 type of 5):
    Type                     Number              Total
    5                          1                     1
    4+1                        2                    50
    3+2                        2                   200
    3+1+1                      3                   750
    2+2+1                      3                  1500
    2+1+1+1                    4                  5000
    1+1+1+1+1+1                1                  3125,
    adding to a(5)=10626.

Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).

Cf. A000012, A000129, A108368, A374454.

Cf. A145841.

CoefficientList[Series[1/(1-5*x-10*x^2-10*x^3-5*x^4-x^5),{x,0,22}],x] (* Stefano Spezia, Jul 09 2024 *)

a(n) = 5*a(n-1) + 10*a(n-2) + 10*a(n-3) + 5*a(n-4) + a(n-5), n=>5.

a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(5*k,n). - Seiichi Manyama, Aug 03 2024

a(20) corrected by Georg Fischer, Oct 28 2024

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A108369 Coefficients of x/(1+3*x+3*x^2-x^3).

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A377314 a(n) = coefficient of the term that is independent of 2^(1/3) and 2^(2/3) in the expansion of (1 + 2^(1/3) + 2^(2/3))^n.

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A377315 a(n) = coefficient of 2^(1/3) in the expansion of (1 + 2^(1/3) + 2^(2/3))^n.

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A374454 Expansion of o.g.f. 1/(1 - 4*x - 6*x^2 - 4*x^3 - x^4).

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A374455 Expansion of o.g.f. 1/(1 - 5*x - 10*x^2 - 10*x^3 - 5*x^4 - x^5).

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A108369 Coefficients of x/(1+3x+3x^2-x^3).

A374454 Expansion of o.g.f. 1/(1 - 4x - 6x^2 - 4*x^3 - x^4).

A374455 Expansion of o.g.f. 1/(1 - 5x - 10x^2 - 10x^3 - 5x^4 - x^5).