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.

A385498 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k).

Original entry on oeis.org

1, 6, 48, 408, 3564, 31626, 283548, 2560872, 23255964, 212101176, 1941110628, 17815257048, 163896843300, 1510891524252, 13952756564424, 129048895061208, 1195191116753436, 11082661017288264, 102877353868090080, 955912961224763232, 8889969049985302464
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2025

Keywords

Crossrefs

Cf. A100192, A385004, A386699.
Cf. A005810, A066381, A371814, A386701.

Programs

  • Mathematica
    Table[(81/8)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n, k));

Formula

a(n) = [x^n] 1/((1-3*x) * (1-x)^(3*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(3*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(139*n^3 - 366*n^2 + 143*n + 132)*a(n) = (588665*n^6 - 2281011*n^5 + 2262209*n^4 + 1245939*n^3 - 3359986*n^2 + 1877400*n - 322560)*a(n-1) - 648*(2*n - 3)*(4*n - 7)*(4*n - 5)*(139*n^3 + 51*n^2 - 172*n + 48)*a(n-2).
a(n) ~ 2^(8*n + 1/2) / (sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
G.f.: g/((3-2*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} 3^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*(12-6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A386700 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n,k).

Original entry on oeis.org

1, 0, 6, 30, 186, 1140, 7116, 44856, 285066, 1823232, 11721726, 75683718, 490429224, 3187723344, 20774505408, 135699314640, 888177411018, 5823660624408, 38245666664994, 251528316024042, 1656338630258826, 10919849458481028, 72068276593960884, 476093333668519872
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2025

Keywords

Crossrefs

Cf. A122803, A386701, A386702.
Cf. A005809, A066380, A165817, A371813, A385004.

Programs

  • Mathematica
    Table[(-8/9)^n - Binomial[3*n, n]*(-1 + Hypergeometric2F1[1, -2*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-3)^(n-k)*binomial(3*n, k));

Formula

a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 18*n*(2*n - 1)*(55*n^2 - 175*n + 138)*a(n) = (11605*n^4 - 49410*n^3 + 74243*n^2 - 46014*n + 9720)*a(n-1) + 24*(3*n - 5)*(3*n - 4)*(55*n^2 - 65*n + 18)*a(n-2).
a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(2*n)). (End)
G.f.: g/((-2+3*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} (-2)^k * 3^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - 6*x*g^2*(-1+g)) where g = 1+x*g^3 is the g.f. of A001764. - 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

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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