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 1-4 of 4 results.

A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).

Original entry on oeis.org

1, 3, 16, 93, 562, 3473, 21778, 137980, 880970, 5658537, 36519556, 236618693, 1538132224, 10026362492, 65513177704, 428957009288, 2813768603466, 18486790962201, 121634649321208, 801330506737399, 5285305708097522, 34896814868837161, 230631268849574378
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066380, A160906, A385823, A386006, A387007, A387008.
Cf. A114121, A387037.
Cf. A001764, A047099, A165817.

Programs

  • Magma
    [&+[Binomial(3*n-1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    Table[Sum[Binomial[3*n-1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*n-1, k));

Formula

a(n) = [x^n] (1+x)^(3*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-1,k) * binomial(3*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-2,n-k).
G.f.: 1/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+6)*(3*n-4)*(3*n-5)*a(n-2)-(295*n^3-451*n^2-234*n+360)*a(n-1)+2*n*(5*n+1)*(2*n-3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n-1)). - Vaclav Kotesovec, Aug 27 2025

A387034 a(n) = Sum_{k=0..n} binomial(4*n-4,k).

Original entry on oeis.org

1, 1, 11, 93, 794, 6885, 60460, 536155, 4791323, 43081973, 389329652, 3533047572, 32174057272, 293874981603, 2691171713924, 24700051833634, 227150464141969, 2092620625940629, 19308393192688804, 178406554524801820, 1650535921328322392, 15287533448476027572
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066381, A386811, A387009, A387010, A387011, A387035, A387036, A387037.
Cf. A002293, A262977.

Programs

  • Magma
    [&+[Binomial(4*n-4, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[Binomial[4*n-4,k], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Aug 20 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n-4, k));

Formula

a(n) = [x^n] (1+x)^(4*n-4)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-4) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-4,k) * binomial(4*n-k-5,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-5,n-k).
G.f.: 1/(g^3 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-11)*(2*n-5)*(4*n-9)*(44*n^3-122*n^2+18*n+105)*a(n-2)-8*(3784*n^6-37684*n^5+141548*n^4-238406*n^3+145758*n^2+37290*n-51975)*a(n-1)+3*n*(3*n-5)*(3*n-7)*(44*n^3-254*n^2+394*n-79)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n - 17/2) / (sqrt(Pi*n) * 3^(3*n - 9/2)). - Vaclav Kotesovec, Aug 20 2025

A387035 a(n) = Sum_{k=0..n} binomial(4*n-3,k).

Original entry on oeis.org

1, 2, 16, 130, 1093, 9402, 82160, 726206, 6474541, 58115146, 524472448, 4754293704, 43257431931, 394821713910, 3613377083248, 33146854168628, 304692552429413, 2805871076597738, 25880523571338272, 239058748663208600, 2211058130414688244, 20474163633488699944
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387036, A387037.
Cf. A002293, A262977.

Programs

  • Magma
    [&+[Binomial(4*n-3, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[Binomial[4*n-3,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n-3, k));

Formula

a(n) = [x^n] (1+x)^(4*n-3)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-3,k) * binomial(4*n-k-4,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-4,n-k).
G.f.: 1/(g^2 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-7)*(2*n-5)*(4*n-9)*(22*n^3-50*n^2+5*n+30)*a(n-2) -8*(1892*n^6-16004*n^5+51038*n^4-73470*n^3+39874*n^2+6165*n-9450)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^3-116*n^2+171*n-47)*a(n) = 0. - Georg Fischer, Aug 17 2025

A387036 a(n) = Sum_{k=0..n} binomial(4*n-2,k).

Original entry on oeis.org

1, 3, 22, 176, 1471, 12616, 110056, 971712, 8656937, 77663192, 700614760, 6349125440, 57754842117, 527046644056, 4822774262296, 44235726874816, 406582639811581, 3743845040832376, 34529632747211560, 318931047174438720, 2949641596923575548, 27312107861301870368
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387035, A387037.
Cf. A002293, A262977.

Programs

  • Magma
    [&+[Binomial(4*n-2, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[Binomial[4*n-2,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n-2, k));

Formula

a(n) = [x^n] (1+x)^(4*n-2)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-2,k) * binomial(4*n-k-3,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-3,n-k).
G.f.: 1/(g * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-7)*(2*n-3)*(4*n-9)*(22*n^2-17*n-15)*a(n-2) -8*(1892*n^5-11274*n^4+23326*n^3-18132*n^2+1323*n+2835)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^2-61*n+24)*a(n) = 0. - Georg Fischer, Aug 17 2025
Showing 1-4 of 4 results.

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