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.

Previous Showing 11-16 of 16 results.

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

A385497 a(n) = Sum_{k=0..n} binomial(6*n+1,k).

Original entry on oeis.org

1, 8, 92, 1160, 15276, 206368, 2835200, 39419864, 553000876, 7811733392, 110962066532, 1583318009160, 22677731944032, 325849065291056, 4694837606889424, 67803714186207280, 981265566082447276, 14227018304102548368, 206608052310739404392, 3004777578508008253808
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2025

Keywords

Links

Crossrefs

Cf. A000302, A160906, A386811, A386812.
Cf. A004355, A079590, A079679, A226705.
Cf. A002295, A385719.

Programs

  • Magma
    [&+[Binomial(6*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 18 2025
  • Mathematica
    Table[Sum[Binomial[6*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 18 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(6*n+1, k));

Formula

a(n) = [x^n] (1+x)^(6*n+1)/(1-x).
a(n) = [x^n] 1/((1-x)^(5*n+1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(6*n-k,n-k).
G.f.: 1/(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g^2/((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: B(x)^2/(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.
a(n) ~ 2^(6*n-1) * 3^(6*n + 3/2) / (sqrt(Pi*n) * 5^(5*n + 1/2)). - Vaclav Kotesovec, Aug 19 2025
D-finite with recurrence +5*n*(5*n-3) *(25275337086729240289198339046875*n +471647298106881091699147254457046) *(5*n-1)*(5*n-4)*(5*n-2)*a(n) +(78985428396028875903744809521484375*n^6 -559942234844855804767211877804090453801*n^5 +3587636672285250929619857349305543417315*n^4 -10153151347942687598200945831585305558855*n^3 +14794114656715293872778407292185015920550*n^2 -10846691360081598422810600143797325763664*n +3179147242764665659301361496311050364480)*a(n-1) +40*(916451705547792050816664342989042382392*n^6 -15754440652132350078674083937326518806004*n^5 +117614110896134855700514819789186651267682*n^4 -471111363407608954402735569277858473721059*n^3 +1053743992048348087929158710510276422876431*n^2 -1242809524683997363700671579060256757555078*n +603414490131980309336751304501155726403152) *a(n-2) +3072*(-950768355029313182341332806167821761828*n^6 +17097100921628721474237101055297828968024*n^5 -128090998271831890487248970509140383514230*n^4 +509544263618626898681417576914870842148685*n^3 -1132270964907780344616429736070172799129247*n^2 +1330655887974191637410201798934319046990726*n -645481184978535641217111809931780144149880) *a(n-3) +884736*(3*n-11) *(6*n-17) *(61801507754400081418308631750717123*n -123657551673181017806623428016627104) *(6*n-19)*(3*n-10)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Aug 26 2025

A385639 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 7, 69, 748, 8485, 98847, 1171884, 14066808, 170421669, 2079531685, 25520363869, 314653207128, 3894577133356, 48362609654548, 602248101550920, 7517853111444528, 94044248726758821, 1178641094940246897, 14796230460187072719, 186022053254555479500, 2341837809478393341885
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Cf. A098430, A383716, A386811.
Cf. A178792, A244038, A386895.

Programs

  • Mathematica
    Table[Sum[Binomial[4*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k, n-k));

Formula

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-4*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
a(n) ~ sqrt((187 - 3*sqrt(17)) / (17*Pi*n)) * (51*sqrt(17) - 107)^n / 2^(3*n + 3/2). - Vaclav Kotesovec, Aug 07 2025
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