A371458 -id:A371458 - cp's OEIS frontend

A371456 Expansion of 1/(1 - x/(1 - 9*x^2)^(1/3)).

1, 1, 1, 4, 7, 28, 58, 223, 505, 1876, 4498, 16255, 40576, 143422, 368965, 1280830, 3373225, 11536309, 30958240, 104559082, 284934754, 952183048, 2628211291, 8703329266, 24283705558, 79785964555, 224677646416, 733160045533, 2081054132179, 6750196280983

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A362206, A371458.

Cf. A373543.

A371456 := proc(n)
    add(9^k*binomial((n+k)/3-1,k),k=0..floor(n/2)) ;
end proc:
seq(A371456(n),n=0..70) ; # R. J. Mathar, Jun 07 2024

a(n) = sum(k=0, n\2, 9^k*binomial((n+k)/3-1, k));

a(n) = Sum_{k=0..floor(n/2)} 9^k * binomial((n+k)/3-1,k).
D-finite with recurrence -(n-1)*(n-2)*(n-8)*a(n) +3*(9*n^3-123*n^2+490*n-616)*a(n-2) +(n-1)*(n-2)*(n-8)*a(n-3) +9*(-27*n^3+441*n^2-2318*n+3984)*a(n-4) +6*(-3*n^3+45*n^2-206*n+284)*a(n-5) +81*(3*n-20)*(n-6)*(3*n-19)*a(n-6) +9*(3*n-20)*(n-6)*(3*n-19)*a(n-7)=0. - R. J. Mathar, Jun 07 2024
a(n) == 1 (mod 3). - Seiichi Manyama, Jun 11 2024

A373510 Expansion of 1/(1 - x/(1 - 25*x^5)^(1/5)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 11, 16, 21, 26, 106, 211, 341, 496, 676, 2256, 4611, 7866, 12146, 17576, 51781, 106761, 188266, 302671, 456976, 1236306, 2552661, 4602416, 7620071, 11881376, 30218956, 62278561, 114056566, 193134346, 308915776, 749942856, 1540351961

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A066872, A098615, A371458, A373511.

a(n) = sum(k=0, n\5, 25^k*binomial(n/5-1, k));

a(5*n) = 26^(n-1) for n > 0.
a(n) = Sum_{k=0..floor(n/5)} 25^k * binomial(n/5-1,k).
a(n) == 1 (mod 5).

A373278 Expansion of 1 / ( (1 - 9x^3) (1 - x/(1 - 9*x^3)^(1/3)) ).

Original entry on oeis.org

1, 1, 1, 10, 13, 16, 100, 148, 205, 1000, 1606, 2410, 10000, 17005, 27070, 100000, 177421, 295648, 1000000, 1833178, 3168538, 10000000, 18811948, 33503020, 100000000, 192080866, 350707345, 1000000000, 1953820210, 3642942040, 10000000000, 19815499120, 37611477133

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A000079, A100095, A373583, A373621.

Cf. A011557, A371458.

a(n) = sum(k=0, n\3, 9^k*binomial(n/3, k));

a(3*n) = 10^n for n >= 0.
a(n) = Sum_{k=0..floor(n/3)} 9^k * binomial(n/3,k).
a(n) == 1 (mod 3).
D-finite with recurrence (n-1)*(n-2)*a(n) +2*(-14*n^2+69*n-91)*a(n-3) +9*(n-3)*(29*n-114)*a(n-6) -810*(n-3)*(n-6)*a(n-9)=0. - R. J. Mathar, Jun 21 2024

A373511 Expansion of 1/(1 - x/(1 - 49*x^7)^(1/7)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 8, 15, 22, 29, 36, 43, 50, 253, 505, 806, 1156, 1555, 2003, 2500, 9906, 20105, 33440, 50254, 70890, 95691, 125000, 423270, 861190, 1467915, 2275001, 3316405, 4628485, 6250000, 18944976, 38420768, 66538494, 105430585, 157517592, 225524993, 312500000

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A066872, A098615, A371458, A373510.

a(n) = sum(k=0, n\7, 49^k*binomial(n/7-1, k));

a(7*n) = 50^(n-1) for n > 0.
a(n) = Sum_{k=0..floor(n/7)} 49^k * binomial(n/7-1,k).
a(n) == 1 (mod 7).

A373544 Expansion of 1/(1 - x/(1 - 9*x^3)^(2/3)).

Original entry on oeis.org

1, 1, 1, 1, 7, 13, 19, 70, 157, 280, 799, 1894, 3781, 9646, 23080, 49159, 119203, 283972, 627760, 1487095, 3518617, 7945561, 18620746, 43801447, 100117099, 233475802, 546859390, 1258634440, 2928668632, 6839770279, 15804569341, 36739434904, 85640217781, 198337427839

Offset: 0

Seiichi Manyama, Jun 09 2024

nonn

Cf. A362210, A373543.

Cf. A371458.

a(n) = sum(k=0, n\3, 9^k*binomial(2*n/3-k-1, k));

a(n) = Sum_{k=0..floor(n/3)} 9^k * binomial(2*n/3-k-1,k).

a(n) == 1 (mod 3).

A386721 Expansion of e.g.f. exp(x/(1 - 9*x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 73, 361, 1081, 93241, 912241, 4907953, 476295121, 7244922961, 58360393081, 6211842488281, 130899060524233, 1435239754046281, 164948740478252641, 4498516738183799521, 63300797606830713121, 7772118657831401082913, 262261735708117281036841

Offset: 0

Seiichi Manyama, Sep 03 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Vaclav Kotesovec, Graph - the asymptotic ratio (20000 terms)

Cf. A189054, A371458, A373517.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1 - 9*x^3)^(1/3)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Sep 03 2025

nmax = 20; CoefficientList[Series[E^(x/(1 - 9*x^3)^(1/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *)

a(n) = n!*sum(k=0, n\3, 9^k*binomial(n/3-1, k)/(n-3*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} 9^k * binomial(n/3-1,k)/(n-3*k)!.
a(n) == 1 mod 72.
From Vaclav Kotesovec, Sep 03 2025: (Start)
a(n) = (36*n^3 - 324*n^2 + 1008*n - 1079)*a(n-3) - 162*(n-6)*(n-5)*(n-4)*(n-3)*(3*n^2 - 27*n + 64)*a(n-6) + 2916*(n-9)*(n-8)*(n-7)*(n-6)^3*(n-5)*(n-4)*(n-3)*a(n-9) - 6561*(n-12)*(n-11)*(n-10)*(n-9)^2*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*a(n-12).
a(n) ~ 3^(2*n/3 - 1/4) * exp(4*3^(-3/2)*n^(1/4) - n) * n^(n - 3/8) / 2. (End)

cp's OEIS Frontend

A371456 Expansion of 1/(1 - x/(1 - 9*x^2)^(1/3)).

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A373510 Expansion of 1/(1 - x/(1 - 25*x^5)^(1/5)).

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A373278 Expansion of 1 / ( (1 - 9*x^3) * (1 - x/(1 - 9*x^3)^(1/3)) ).

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A373511 Expansion of 1/(1 - x/(1 - 49*x^7)^(1/7)).

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A373544 Expansion of 1/(1 - x/(1 - 9*x^3)^(2/3)).

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PARI

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A386721 Expansion of e.g.f. exp(x/(1 - 9*x^3)^(1/3)).

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A373278 Expansion of 1 / ( (1 - 9x^3) (1 - x/(1 - 9*x^3)^(1/3)) ).