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 1-5 of 5 results.

A385605 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k).

Original entry on oeis.org

1, 7, 58, 502, 4436, 39687, 358024, 3249288, 29624796, 271080124, 2487835678, 22888216006, 211010997716, 1948830506578, 18026768864736, 166976297995452, 1548523206590364, 14376415735219572, 133599985919343400, 1242638966005222648, 11567295503871866536
Offset: 0

Views

Author

Seiichi Manyama, Aug 03 2025

Keywords

Crossrefs

Cf. A006256, A141223, A385632.
Cf. A005810, A078995, A147855, A386811.
Cf. A385498.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k));

Formula

a(n) = [x^n] 1/((1-3*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(3*n+k,k).
a(n) = 3^(4*n+1)*2^(-3*n-1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((3-2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 14 2025
G.f.: B(x)^2/(1 + (B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(12-5*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 16 2025

A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).

Original entry on oeis.org

1, 7, 69, 733, 8061, 90462, 1028871, 11814376, 136643085, 1589311381, 18569375114, 217773347502, 2561944357311, 30219704365104, 357278540928168, 4232449819704768, 50227362114232109, 596988743410929087, 7105534815529752831, 84678089652554263155, 1010268312800732117946
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2025

Keywords

Crossrefs

Cf. A000244, A100192, A385004, A385498.
Cf. A163455, A371739, A386702.
Cf. A002294.

Programs

  • Mathematica
    Table[(243/16)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(5*n, k));

Formula

a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 128*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(3052*n^4 - 15114*n^3 + 26432*n^2 - 18693*n + 4131)*a(n) = 8*(42807352*n^8 - 285737492*n^7 + 758983420*n^6 - 1002945218*n^5 + 644348866*n^4 - 111879380*n^3 - 84004497*n^2 + 44187381*n - 5806080)*a(n-1) - 1215*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(3052*n^4 - 2906*n^3 - 598*n^2 + 1037*n - 192)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi*n) * 2^(8*n + 1/2)). (End)
G.f.: g/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(15-8*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A385004 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k).

Original entry on oeis.org

1, 5, 31, 200, 1311, 8665, 57556, 383556, 2561871, 17140007, 114819351, 769925568, 5166845124, 34696155564, 233113911208, 1566926561740, 10536427052463, 70872688450083, 476854924775869, 3209222876463192, 21602639249766951, 145444151677134153, 979397744169608784
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2025

Keywords

Crossrefs

Cf. A000244, A100192, A385498, A386699.
Cf. A005809, A066380, A165817, A371813, A386700.
Cf. A001764, A386617.

Programs

  • Mathematica
    Table[(27/4)^n - Binomial[3*n, n] * (-1 + Hypergeometric2F1[1, -2*n, 1 + n, -1/2]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n, k));

Formula

a(n) = [x^n] 1/((1-3*x) * (1-x)^(2*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 8*n*(2*n - 1)*(15*n - 23)*a(n) = 6*(540*n^3 - 1503*n^2 + 1239*n - 320)*a(n-1) - 81*(3*n - 5)*(3*n - 4)*(15*n - 8)*a(n-2).
a(n) ~ 3^(3*n) / 2^(2*n+1) * (1 + 5/(3*sqrt(3*Pi*n))). (End)
G.f.: g/(3-2*g)^2 where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(9-4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

A386701 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n,k).

Original entry on oeis.org

1, 1, 13, 103, 869, 7476, 65323, 577242, 5144949, 46167196, 416527828, 3774785983, 34336862435, 313330665532, 2866982877954, 26294890918308, 241665561294741, 2225104901535564, 20520648006149980, 189523353219338572, 1752680220372189364, 16227703263403842768
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2025

Keywords

Crossrefs

Cf. A386700, A386702.
Cf. A066381, A262977, A371814, A385498.

Programs

  • Mathematica
    Table[(-16/27)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-3)^(n-k)*binomial(4*n, k));

Formula

a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(612*n^3 - 2838*n^2 + 4354*n - 2209)*a(n) = 24*(165240*n^6 - 1019628*n^5 + 2493432*n^4 - 3068178*n^3 + 1984652*n^2 - 632900*n + 76545)*a(n-1) + 128*(2*n - 3)*(4*n - 7)*(4*n - 5)*(612*n^3 - 1002*n^2 + 514*n - 81)*a(n-2).
a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). (End)
G.f.: g/((-2+3*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-8+9*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 2, 16, 128, 1068, 9142, 79612, 701864, 6244892, 55962920, 504375396, 4567003520, 41513817444, 378596616452, 3462411408136, 31742042431048, 291616814436124, 2684123914512280, 24746511514749280, 228491677484832896, 2112549277665243328
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A072547, A371813.
Cf. A005810, A262977.
Cf. A066381, A262977, A385498, A386701.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(4*n-k-1, n-k));

Formula

a(n) = [x^n] 1/((1+x) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, -n], [1-4*n], -1). - Stefano Spezia, Apr 07 2024
From Vaclav Kotesovec, Apr 07 2024: (Start)
Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(415*n^3 - 1898*n^2 + 2871*n - 1436)*a(n) = (838715*n^6 - 5099533*n^5 + 12225995*n^4 - 14652035*n^3 + 9157250*n^2 - 2799192*n + 322560)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(415*n^3 - 653*n^2 + 320*n - 48)*a(n-2).
a(n) ~ 2^(8*n + 1/2) / (5 * sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n,k). - Seiichi Manyama, Jul 30 2025
G.f.: g/((-1+2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-4+6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025
Showing 1-5 of 5 results.

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