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 1-3 of 3 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

A382332 Expansion of 1/(1 - 4*x/(1-x)^2)^(7/2).

Original entry on oeis.org

1, 14, 154, 1470, 12866, 106078, 837018, 6385262, 47420674, 344553902, 2458367898, 17272647966, 119770278978, 821068784382, 5572735854234, 37490757508302, 250247764120578, 1658681038111566, 10924592141535898, 71541334475749502, 466060971286552642
Offset: 0

Views

Author

Seiichi Manyama, Mar 30 2025

Keywords

Links

Crossrefs

Cf. A110170, A377198, A382274.
Cf. A020918, A377200.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(7/2))); // Vincenzo Librandi, May 12 2025
  • Mathematica
    Table[Sum[(-4)^k* Binomial[-7/2,k]*Binomial[n+k-1, n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n+k-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (7-5*k/n) * (n-k) * a(k).
a(n) = ((7*n+7)*a(n-1) - (7*n-28)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+k-1,n-k).
a(n) = 14*n*hypergeom([9/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025
a(n) ~ 2^(5/4) * (1 + sqrt(2))^(2*n) * n^(5/2) / (15*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 1-3 of 3 results.

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