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.

A377199 Expansion of 1/(1 - 4*x/(1-x))^(5/2).

Original entry on oeis.org

1, 10, 80, 570, 3790, 24062, 147780, 885190, 5199560, 30065870, 171623328, 969151710, 5422863630, 30105497970, 165993714540, 909770119914, 4959840748350, 26912374137150, 145411035749600, 782681600883950, 4198276264607290, 22448626776903450, 119690255236279100
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2024

Keywords

Links

Crossrefs

Cf. A085362, A377197, A377200.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(5/2))); // Vincenzo Librandi, May 10 2025
  • Mathematica
    Table[Sum[(-4)^k*Binomial[-5/2,k]*Binomial[n-1,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, May 10 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-4)^k*binomial(-5/2, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (5-3*k/n) * a(k).
a(n) = (2*(3*n+2)*a(n-1) - 5*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n-1,n-k).
a(n) ~ 128 * n^(3/2) * 5^(n - 5/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, Oct 26 2024
a(n) = 10*hypergeom([7/2, 1-n], [2], -4) for n > 0. - Stefano Spezia, May 08 2025

A377198 Expansion of 1/(1 - 4*x/(1-x)^2)^(3/2).

Original entry on oeis.org

1, 6, 42, 278, 1794, 11382, 71338, 443046, 2732034, 16751462, 102235050, 621535158, 3766261506, 22758222294, 137186860842, 825211984710, 4954574749698, 29697908825286, 177746214414634, 1062416305340502, 6342559258130178, 37823152988963126, 225328426205608362
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2024

Keywords

Links

Crossrefs

Cf. A002457, A377197.
Cf. A377194.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(3/2))); // Vincenzo Librandi, May 11 2025
  • Mathematica
    Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n+k-1,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, May 11 2025 *)
  • PARI
    a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n+k-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (3-k/n) * (n-k) * a(k).
a(n) = ((7*n-1)*a(n-1) - (7*n-20)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+k-1,n-k).
a(n) ~ 2^(1/4) * sqrt(n) * (1 + sqrt(2))^(2*n) / sqrt(Pi). - Vaclav Kotesovec, May 03 2025
a(n) = 6*n*hypergeom([5/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, May 08 2025

A377200 Expansion of 1/(1 - 4*x/(1-x))^(7/2).

Original entry on oeis.org

1, 14, 140, 1190, 9170, 66122, 454328, 3009050, 19359620, 121664410, 749879508, 4546925922, 27188341530, 160624341990, 939009926520, 5438826037974, 31244200818306, 178173537480330, 1009366349014100, 5684102310204850, 31836106214747590, 177430881586034110
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2024

Keywords

Links

Crossrefs

Cf. A085362, A377197, A377199.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(7/2))); // Vincenzo Librandi, May 11 2025
  • Mathematica
    Table[Sum[(-4)^k*Binomial[-7/2,k]*Binomial[n-1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (7-5*k/n) * a(k).
a(n) = (2*(3*n+4)*a(n-1) - 5*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n-1,n-k).
a(n) ~ 1024 * 5^(n - 9/2) * n^(5/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, May 03 2025
a(n) = 14*hypergeom([9/2, 1-n], [2], -4) for n > 0. - Stefano Spezia, May 08 2025

A377204 Expansion of 1/(1 - 4*x^2/(1-x))^(3/2).

Original entry on oeis.org

1, 0, 6, 6, 36, 66, 236, 546, 1626, 4106, 11388, 29646, 79838, 209718, 557328, 1465970, 3869448, 10166370, 26726080, 70092570, 183756378, 481048010, 1258494768, 3289100958, 8590288128, 22418099982, 58467588768, 152388145382, 396954437202, 1033452111702, 2689186662552
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2024

Keywords

Links

Crossrefs

Cf. A377197, A377213.
Cf. A026585, A377215.
Cf. A377186.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2/(1-x))^(3/2))); // Vincenzo Librandi, May 08 2025
  • Mathematica
    Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-k-1,n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 08 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, (2*k+1)*binomial(2*k, k)*binomial(n-k-1, n-2*k));

Formula

a(n) = (2*(n-1)*a(n-1) + (3*n+6)*a(n-2) - 2*(2*n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(n-k-1,n-2*k).
a(n) ~ sqrt(n) * 2^(3*n - 1/2) / (17^(3/4) * sqrt(Pi) * (sqrt(17) - 1)^(n - 3/2)). - Vaclav Kotesovec, May 03 2025

A377213 Expansion of 1/(1 - 4*x^3/(1-x))^(3/2).

Original entry on oeis.org

1, 0, 0, 6, 6, 6, 36, 66, 96, 266, 576, 1026, 2246, 4866, 9516, 19598, 41286, 83526, 170048, 351378, 716850, 1458098, 2984028, 6087270, 12380900, 25224222, 51356400, 104380510, 212164362, 431148222, 875353220, 1776567762, 3604752672, 7310374010, 14819370480, 30033014994
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2024

Keywords

Links

Crossrefs

Cf. A377197, A377204.
Cf. A360309, A377216.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^3/(1-x))^(3/2))); // Vincenzo Librandi, May 08 2025
  • Mathematica
    Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-2*k-1,n-3*k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 08 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, (2*k+1)*binomial(2*k, k)*binomial(n-2*k-1, n-3*k));

Formula

a(n) = (2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n+3)*a(n-3) - 2*(2*n-2)*a(n-4))/n for n > 3.
a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(n-2*k-1,n-3*k).
a(n) ~ sqrt(n) * 2^(n-2) / sqrt(Pi). - Vaclav Kotesovec, May 03 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.