cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A371458 Expansion of 1/(1 - x/(1 - 9*x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 10, 31, 61, 100, 274, 565, 1000, 2551, 5380, 10000, 24376, 52018, 100000, 236389, 507706, 1000000, 2313346, 4986178, 10000000, 22773334, 49180165, 100000000, 225092416, 486575935, 1000000000, 2231117230, 4824998773, 10000000000
Offset: 0

Views

Author

Seiichi Manyama, Jun 07 2024

Keywords

Crossrefs

Cf. A362206, A371456.
Cf. A066872, A098615, A373510, A373511.

Programs

  • Maple
    A371458 := proc(n)
    add(9^k*binomial(n/3-1,k),k=0..floor(n/3)) ;
end proc:
seq(A371458(n),n=0..70) ; # R. J. Mathar, Jun 07 2024
  • PARI
    a(n) = sum(k=0, n\3, 9^k*binomial(n/3-1, k));

Formula

a(3*n) = 10^(n-1) for n > 0.
a(n) = Sum_{k=0..floor(n/3)} 9^k * binomial(n/3-1,k).
D-finite with recurrence (n-1)*(n-2)*a(n) +4*(-7*n^2+48*n-86)*a(n-3) +9*(29*n-141)*(n-6)*a(n-6) -810*(n-6)*(n-9)*a(n-9)=0. - R. J. Mathar, Jun 07 2024
a(n) == 1 (mod 3). - Seiichi Manyama, Jun 11 2024

A362210 Expansion of 1/(1 - x/(1-9*x)^(2/3)).

Original entry on oeis.org

1, 1, 7, 58, 505, 4498, 40576, 368965, 3373225, 30958240, 284934754, 2628211291, 24283705558, 224677646416, 2081054132179, 19293026227024, 178996540057615, 1661743445778403, 15435351753092176, 143439377236572826, 1333496145331028230
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2023

Keywords

Crossrefs

Cf. A362206.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-x/(1-9*x)^(2/3)))

Formula

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

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jun 07 2024

Keywords

Crossrefs

Cf. A362206, A371458.
Cf. A373543.

Programs

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

Formula

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

A369627 Expansion of 1/(1 - x^2/(1-9*x)^(1/3)).

Original entry on oeis.org

1, 0, 1, 3, 19, 132, 991, 7740, 62020, 505857, 4180132, 34889514, 293518072, 2485191753, 21153817090, 180865139538, 1552289627872, 13366436688402, 115425148203235, 999256943147094, 8670047414816233, 75375298322580081, 656465004512563546
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2024

Keywords

Links

Crossrefs

Cf. A362206, A369940.
Cf. A104625.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - x^2/(1-9*x)^(1/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 19 2024 *)
Flatten[{{1, 0, 1, 3, 19, 132}, RecurrenceTable[{9 (-12 + n) (-19 + 3 n) (-14 + 3 n) a[-8 + n] - 6 (-628 + 368 n - 63 n^2 + 3 n^3) a[-7 + n] + (-13 + n) (-4 + n) (-2 + n) a[-6 + n] + 81 (-12 + n) (-19 + 3 n) (-14 + 3 n) a[-3 + n] - 9 (-6960 + 3662 n - 585 n^2 + 27 n^3) a[-2 + n] + 3 (-14 + 3 n) (112 - 47 n + 3 n^2) a[-1 + n] - (-13 + n) (-4 + n) (-2 + n) a[n] == 0, a[6] == 991, a[7] == 7740, a[8] == 62020, a[9] == 505857, a[10] == 4180132, a[11] == 34889514, a[12] == 293518072, a[13] == 2485191753}, a, {n, 6, 20}]}] (* Vaclav Kotesovec, Feb 19 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-9*x)^(1/3)))

Formula

a(n) = Sum_{k=0..floor(n/2)} 9^(n-2*k) * binomial(n-1-5*k/3,n-2*k).
a(n) ~ (r-9)^(4/3) * r^(5/3) * r^n / (2*r-15), where r = 9.0000169349284790514638157821699098461789951085871459872133... = is the largest real root of the equation r^5*(r-9) = 1. - Vaclav Kotesovec, Feb 19 2024

A369940 Expansion of 1/(1 - x^3/(1-9*x)^(1/3)).

Original entry on oeis.org

1, 0, 0, 1, 3, 18, 127, 951, 7416, 59329, 483147, 3986415, 33224338, 279121233, 2360156580, 20063973502, 171337660872, 1468794800925, 12633200032942, 108974515627170, 942420040015635, 8168578134973084, 70945593205544931, 617294050087428540
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2024

Keywords

Links

Crossrefs

Cf. A362206, A369627.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - x^3/(1-9*x)^(1/3)), {x, 0, 25}], x] (* Vaclav Kotesovec, Feb 20 2024 *)
Join[{1, 0, 0, 1, 3, 18, 127, 951, 7416}, RecurrenceTable[{9 (-18 + n) (-25 + 3 n) (-17 + 3 n) a[-11 + n] - 18 (-566 + 243 n - 30 n^2 + n^3) a[-10 + n] + (-19 + n) (-6 + n) (-3 + n) a[-9 + n] + 81 (-18 + n) (-25 + 3 n) (-17 + 3 n) a[-3 + n] - 9 (-17838 + 7067 n - 828 n^2 + 27 n^3) a[-2 + n] + 9 (-1474 + 675 n - 88 n^2 + 3 n^3) a[-1 + n] - (-19 + n) (-6 + n) (-3 + n) a[n] == 0, a[9] == 59329, a[10] == 483147, a[11] == 3986415, a[12] == 33224338, a[13] == 279121233, a[14] == 2360156580, a[15] == 20063973502, a[16] == 171337660872, a[17] == 1468794800925, a[18] == 12633200032942, a[19] == 108974515627170}, a, {n, 9, 25}]] (* Vaclav Kotesovec, Feb 20 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-x^3/(1-9*x)^(1/3)))

Formula

a(n) = Sum_{k=0..floor(n/3)} 9^(n-3*k) * binomial(n-1-8*k/3,n-3*k).
From Vaclav Kotesovec, Feb 20 2024: (Start)
Recurrence (for n>19): (n-19)*(n-6)*(n-3)*a(n) = 9*(3*n^3 - 88*n^2 + 675*n - 1474)*a(n-1) - 9*(27*n^3 - 828*n^2 + 7067*n - 17838)*a(n-2) + 81*(n - 18)*(3*n - 25)*(3*n - 17)*a(n-3) + (n - 19)*(n-6)*(n-3)*a(n-9) - 18*(n^3 - 30*n^2 + 243*n - 566)*a(n-10) + 9*(n - 18)*(3*n - 25)*(3*n - 17)*a(n-11).
a(n) ~ (r-9)^(4/3) * r^(8/3) * r^n / (3*(r-8)), where r = 9.00000002323057264572143212814577340192663286000333917759... is the root of the equation (r-9)*r^8 = 1. (End)

A382543 Expansion of 1/(1 - x/(1 - 9*x)^(1/3))^2.

Original entry on oeis.org

1, 2, 9, 58, 428, 3360, 27295, 226538, 1907889, 16239034, 139326959, 1202856930, 10436521180, 90920984306, 794767853334, 6967126281976, 61224158085137, 539141091531558, 4756357637006941, 42028309478725094, 371898032568193530, 3294977494088601508
Offset: 0

Views

Author

Seiichi Manyama, Mar 31 2025

Keywords

Links

Crossrefs

Cf. A362206, A382544.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 9*x)^(1/3))^2)); // Vincenzo Librandi, May 12 2025
  • Mathematica
    Table[Sum[9^(n-k)* (k+1)* Binomial[n-2*k/3-1, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, May 12 2025 *)
  • PARI
    a(n) = sum(k=0, n, 9^(n-k)*(k+1)*binomial(n-2*k/3-1, n-k));

Formula

a(n) = Sum_{k=0..n} 9^(n-k) * (k+1) * binomial(n-2*k/3-1,n-k).
D-finite with recurrence (n-1)*(n-2)*a(n) -3*(n-2)*(11*n-29)*a(n-1) +135*(n-3)*(3*n-10)*a(n-2) +(-2188*n^2+17739*n-35909)*a(n-3) +3*(1466*n^2-14601*n+36292)*a(n-4) +27*(-7*n^2+37*n-42)*a(n-5) +54*(3*n-10)*(3*n-14)*a(n-6)=0. - R. J. Mathar, Apr 02 2025

A382544 Expansion of 1/(1 - x/(1 - 9*x)^(1/3))^3.

Original entry on oeis.org

1, 3, 15, 100, 753, 6006, 49456, 415422, 3536802, 30404161, 263271639, 2292524970, 20052238465, 176029542285, 1549916592645, 13681091072620, 121020187476717, 1072477163769417, 9519299301332377, 84609930915003882, 752947626436806021, 6707715814093174588
Offset: 0

Views

Author

Seiichi Manyama, Mar 31 2025

Keywords

Links

Crossrefs

Cf. A362206, A382543.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 25);f := x / (1 - 9*x)^(1/3);G := 1 / (1 - f)^3;Coefficients(G); // Vincenzo Librandi, Mar 31 2025
  • Mathematica
    Table[Sum[9^(n-k)*Binomial[k+2,2]* Binomial [n-2*k/3-1, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Mar 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 9^(n-k)*binomial(k+2, 2)*binomial(n-2*k/3-1, n-k));

Formula

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

A382516 Expansion of 1/(1 - x/(1 - 9*x)^(4/3)).

Original entry on oeis.org

1, 1, 13, 151, 1693, 18688, 204631, 2230498, 24246229, 263112874, 2852058448, 30892668295, 334454025715, 3619669508056, 39164977065622, 423695451762664, 4583082589819489, 49570596449054509, 536121822834121354, 5798064369702626227, 62702959640721355228
Offset: 0

Views

Author

Seiichi Manyama, Mar 30 2025

Keywords

Crossrefs

Cf. A362206, A362210, A382517.
Cf. A004991.

Programs

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

Formula

a(n) = Sum_{k=0..n} 9^(n-k) * binomial(n+k/3-1,n-k).
D-finite with recurrence (n-1)*(n-2)*a(n) -3*(n-2)*(17*n-35)*a(n-1) +27*(39*n^2-197*n+252)*a(n-2) +2*(-5468*n^2+32199*n-46873)*a(n-3) +6*(9115*n^2-56514*n+77702)*a(n-4) +54*(-1094*n^2-359*n+28901)*a(n-5) +54*(-9846*n^2+134559*n-449254)*a(n-6) +177147*(3*n-19)*(3*n-20)*a(n-7)=0. - R. J. Mathar, Mar 31 2025

A382517 Expansion of 1/(1 - x/(1 - 9*x)^(5/3)).

Original entry on oeis.org

1, 1, 16, 211, 2611, 31426, 373099, 4397527, 51623530, 604629688, 7072089076, 82652922457, 965513250832, 11275328397061, 131649767277064, 1536953772789256, 17941954844917198, 209439428952580837, 2444747948094707815, 28536537876362681194, 333091044353156790346
Offset: 0

Views

Author

Seiichi Manyama, Mar 30 2025

Keywords

Crossrefs

Cf. A362206, A362210, A382516.
Cf. A004992.

Programs

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

Formula

a(n) = Sum_{k=0..n} 9^(n-k) * binomial(n+2*k/3-1,n-k).
D-finite with recurrence (n-1)*(n-2)*a(n) -3*(19*n-37)*(n-2)*a(n-1) +27*(49*n^2-233*n+278)*a(n-2) +2*(-7655*n^2+39732*n-47656)*a(n-3) +3*(25519*n^2-98445*n-28306)*a(n-4) +54*(2552*n^2-69623*n+281314)*a(n-5) +27*(-137799*n^2+1870137*n-6193006)*a(n-6) +177147*(99*n^2-1323*n+4418)*a(n-7) -3188646*(3*n-20)*(3*n-22)*a(n-8)=0. - R. J. Mathar, Apr 02 2025
Showing 1-9 of 9 results.

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