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 1-6 of 6 results.

A387010 a(n) = Sum_{k=0..n} binomial(4*n+3,k).

Original entry on oeis.org

1, 8, 67, 576, 5036, 44552, 397594, 3572224, 32267668, 292750368, 2665685155, 24347665728, 222972599812, 2046626681072, 18823260696452, 173427623923712, 1600383346290116, 14789063407109600, 136838247669241276, 1267571539176770816, 11754134090271100336
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2025

Keywords

Links

Crossrefs

Cf. A066381, A386811, A387009, A387011.
Cf. A002293.

Programs

  • Magma
    [&+[Binomial(4*n+3, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025
  • Mathematica
    Table[Sum[Binomial[4*n+3,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 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+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+2,n-k).
G.f.: g^4/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-1)*(2*n-1)*(4*n-3)*(11*n^2+8*n+1)*a(n-2) -8*(946*n^5-434*n^4-518*n^3+143*n^2+46*n-3)*a(n-1) +3*n*(3*n+1)*(3*n+2)*(11*n^2-14*n+4)*a(n) = 0. - Georg Fischer, Aug 17 2025

A387011 a(n) = Sum_{k=0..n} binomial(4*n+4,k).

Original entry on oeis.org

1, 9, 79, 697, 6196, 55455, 499178, 4514873, 40999516, 373585604, 3414035527, 31278197839, 287191809724, 2642070371194, 24347999094724, 224723513577529, 2076978797223820, 19220104372823340, 178061257422521452, 1651314042800498052, 15328459501269535952
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2025

Keywords

Links

Crossrefs

Cf. A066381, A386811, A387009, A387010.
Cf. A002293.

Programs

  • Magma
    [&+[Binomial(4*n+4, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025
  • Mathematica
    Table[Sum[Binomial[4*n+4,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 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+3,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+3,n-k).
G.f.: g^5/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-3)*(2*n+1)*(4*n-1)*(22*n^2+16*n-3)*a(n-2) -8*(1892*n^5+1024*n^4-1982*n^3-1306*n^2-60*n+27)*a(n-1) +3*(n+1)*(3*n+2)*(3*n+1)*(22*n^2-28*n+3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n + 15/2) / (sqrt(Pi*n) * 3^(3*n + 7/2)). - Vaclav Kotesovec, Aug 18 2025

A387037 a(n) = Sum_{k=0..n} binomial(4*n-1,k).

Original entry on oeis.org

1, 4, 29, 232, 1941, 16664, 145499, 1285624, 11460949, 102875128, 928495764, 8417689504, 76599066579, 699232769512, 6400175653922, 58718827590992, 539822826733397, 4971747032359352, 45863130731297180, 423683961417124576, 3919058645835901556
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387035, A387036.
Cf. A114121, A387033.
Cf. A002293, A262977.

Programs

  • Magma
    [&+[Binomial(4*n-1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[Binomial[4*n-1,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n-1, k));

Formula

a(n) = [x^n] (1+x)^(4*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-1,k) * binomial(4*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-2,n-k).
G.f.: 1/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-5)*(4*n-7)*(2*n-3)*(11*n^2-3*n-3)*a(n-2) -8*(946*n^5-4218*n^4+6512*n^3-3753*n^2+201*n+315)*a(n-1) +3*n*(3*n-2)*(3*n-4)*(11*n^2-25*n+11)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi*n) * 3^(3*n - 3/2)). - Vaclav Kotesovec, Sep 03 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 1-6 of 6 results.

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