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 1-6 of 6 results.

A377197 Expansion of 1/(1 - 4*x/(1-x))^(3/2).

Original entry on oeis.org

1, 6, 36, 206, 1146, 6258, 33728, 180018, 953628, 5021698, 26315676, 137350746, 714455826, 3705635646, 19171860336, 98973407550, 509963556330, 2623133951730, 13472299015580, 69098721151530, 353966981339070, 1811212435206070, 9258333786967920, 47281424213258070
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2024

Keywords

Links

Crossrefs

Cf. A085362, A377199, A377200.
Cf. A002457, A377198.
Cf. A374497.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(3/2))); // Vincenzo Librandi, May 11 2025
  • Mathematica
    Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)
  • PARI
    a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (3-k/n) * a(k).
a(n) = (6*n*a(n-1) - 5*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n-1,n-k).
a(n) ~ 16 * sqrt(n) * 5^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Oct 26 2024
a(n) = 6*hypergeom([5/2, 1-n], [2], -4) for n > 0. - Stefano Spezia, May 08 2025

A387228 Expansion of sqrt((1-x) / (1-5*x)^5).

Original entry on oeis.org

1, 12, 103, 764, 5215, 33728, 210021, 1271504, 7532547, 43859460, 251809701, 1428911652, 8028877233, 44734340784, 247433518875, 1359902816880, 7432212863235, 40416897046740, 218812616979845, 1179889937796900, 6339243523221245, 33947223885549040, 181245459484155935
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2025

Keywords

Links

Crossrefs

Cf. A387229, A387230.
Cf. A085362, A163869.
Cf. A377199.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-5*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025
  • Mathematica
    CoefficientList[Series[Sqrt[(1-x)/(1-5*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-5*x)^5))

Formula

n*a(n) = (6*n+6)*a(n-1) - 5*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * 5^(n - 1/2) * n^(3/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, Aug 23 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

A377215 Expansion of 1/(1 - 4*x^2/(1-x))^(5/2).

Original entry on oeis.org

1, 0, 10, 10, 80, 150, 640, 1550, 5190, 13870, 41912, 115650, 333490, 925970, 2607540, 7220062, 20053700, 55230870, 152005380, 416295350, 1137980678, 3100453710, 8429823180, 22862244210, 61882724100, 167159512794, 450739897980, 1213298505770, 3260824389510
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2024

Keywords

Links

Crossrefs

Cf. A377199, A377216.
Cf. A026585, A377204.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2/(1-x))^(5/2))); // Vincenzo Librandi, May 08 2025
  • Mathematica
    Table[Sum[(-4)^k*Binomial[-5/2,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, (-4)^k*binomial(-5/2, k)*binomial(n-k-1, n-2*k));

Formula

a(n) = (2*(n-1)*a(n-1) + (3*n+14)*a(n-2) - 2*(2*n-1)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..floor(n/2)} (-4)^k * binomial(-5/2,k) * binomial(n-k-1,n-2*k).
a(n) ~ n^(3/2) * 2^(3*n - 1/2) / (3 * 17^(5/4) * sqrt(Pi) * (sqrt(17) - 1)^(n - 5/2)). - Vaclav Kotesovec, May 03 2025

A377216 Expansion of 1/(1 - 4*x^3/(1-x))^(5/2).

Original entry on oeis.org

1, 0, 0, 10, 10, 10, 80, 150, 220, 710, 1620, 2950, 7010, 16110, 32560, 70682, 156810, 329290, 698540, 1507110, 3189742, 6725150, 14279520, 30141730, 63335960, 133297362, 279996460, 586364410, 1227337710, 2566307410, 5355970048, 11166535430, 23259949980, 48389451510
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2024

Keywords

Links

Crossrefs

Cf. A377199, A377215.
Cf. A360309, A377213.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^3/(1-x))^(5/2))); // Vincenzo Librandi, May 10 2025
  • Mathematica
    Table[Sum[(-4)^k*Binomial[-5/2,k]*Binomial[n-2*k-1,n-3*k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 10 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, (-4)^k*binomial(-5/2, 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+9)*a(n-3) - 2*(2*n+2)*a(n-4))/n for n > 3.
a(n) = Sum_{k=0..floor(n/3)} (-4)^k * binomial(-5/2,k) * binomial(n-2*k-1,n-3*k).
a(n) ~ n^(3/2) * 2^(n-3) / (3*sqrt(Pi)). - Vaclav Kotesovec, May 03 2025

A382274 Expansion of 1/(1 - 4*x/(1-x)^2)^(5/2).

Original entry on oeis.org

1, 10, 90, 730, 5570, 40762, 289370, 2007210, 13671170, 91750250, 608294490, 3991833210, 25968131010, 167664187290, 1075453670490, 6858654320970, 43517809896450, 274862176368330, 1728960219827290, 10835520927931930, 67679638209628098, 421442759107879930
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2025

Keywords

Crossrefs

Cf. A110170, A377198, A382332.
Cf. A002802, A377199.

Programs

  • PARI
    a(n) = sum(k=0, n, (-4)^k*binomial(-5/2, k)*binomial(n+k-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (5-3*k/n) * (n-k) * a(k).
a(n) = ((7*n+3)*a(n-1) - (7*n-24)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+k-1,n-k).
a(n) = 10*n*hypergeom([7/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025
a(n) ~ 2^(3/4) * n^(3/2) * (1 + sqrt(2))^(2*n) / (3*sqrt(Pi)). - Vaclav Kotesovec, Apr 13 2025
Showing 1-6 of 6 results.

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

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