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.

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

A371813 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 1, 7, 40, 239, 1461, 9076, 57044, 361711, 2309467, 14827487, 95630272, 619111172, 4021011580, 26187682024, 170960159100, 1118406332655, 7330011083079, 48119501497909, 316354663355384, 2082573599282359, 13726029056757029, 90565080767425744
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A072547, A371814.
Cf. A005809, A066380, A165817, A385004, A386700.

Programs

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

Formula

a(n) = [x^n] 1/((1+x) * (1-x)^(2*n)).
a(n) = binomial(3*n-1, n)*hypergeom([1, -n], [1-3*n], -1). - Stefano Spezia, Apr 07 2024
From Vaclav Kotesovec, Apr 07 2024: (Start)
Recurrence: 8*n*(2*n - 1)*(28*n^2 - 87*n + 67)*a(n) = 2*(1456*n^4 - 6008*n^3 + 8593*n^2 - 4949*n + 960)*a(n-1) + 3*(3*n - 5)*(3*n - 4)*(28*n^2 - 31*n + 8)*a(n-2).
a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n+2)). (End)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n,k). - Seiichi Manyama, Jul 30 2025
G.f.: g/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-1)^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*(-3+4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

A386702 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n,k).

Original entry on oeis.org

1, 2, 24, 248, 2676, 29562, 331956, 3771896, 43242660, 499215146, 5795429764, 67587697872, 791232339756, 9292673328174, 109440405341088, 1291977861163968, 15284200451058724, 181147979395807002, 2150493166839159936, 25567085678133719880, 304368033788893315896
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2025

Keywords

Crossrefs

Cf. A386700, A386701.
Cf. A163455, A371739, A386699.

Programs

  • Mathematica
    Table[(-32/81)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n, k));

Formula

a(n) = [x^n] 1/((1+2*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 648*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(37331*n^4 - 227972*n^3 + 518701*n^2 - 521044*n + 194928)*a(n) = (9143593823*n^8 - 74277961298*n^7 + 253684378280*n^6 - 473415527402*n^5 + 524935655069*n^4 - 351762123620*n^3 + 137998180332*n^2 - 28677229776*n + 2380855680)*a(n-1) + 160*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(37331*n^4 - 78648*n^3 + 58771*n^2 - 18234*n + 1944)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (7 * sqrt(Pi*n) * 2^(8*n - 1/2)). (End)
G.f.: g/((-2+3*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} (-2)^k * 3^(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*(-10+12*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025
Showing 1-4 of 4 results.

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