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.

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

A374511 Expansion of 1/(1 - 4*x - 4*x^2)^(5/2).

Original entry on oeis.org

1, 10, 80, 560, 3640, 22512, 134400, 781440, 4451040, 24939200, 137865728, 753625600, 4080643840, 21916106240, 116877312000, 619457482752, 3265293719040, 17128725519360, 89462514606080, 465434423336960, 2412895587536896, 12468681310412800, 64242981906022400
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2024

Keywords

Crossrefs

Cf. A006139, A374497, A374513.

Programs

  • Mathematica
    a[n_]:=2^(n-3) Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, 2]/3; Array[a,23,0] (* Stefano Spezia, Jul 10 2024 *)
  • PARI
    a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));

Formula

a(0) = 1, a(1) = 10; a(n) = (2*(2*n+3)*a(n-1) + 4*(n+3)*a(n-2))/n.
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = 2^(n-3)*Pochhammer(n+1, 4)*hypergeom([(1-n)/2, -n/2], [3], 2)/3. - Stefano Spezia, Jul 10 2024
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024

A374513 Expansion of 1/(1 - 4*x - 4*x^2)^(7/2).

Original entry on oeis.org

1, 14, 140, 1176, 8904, 62832, 421344, 2718144, 17008992, 103847744, 621292672, 3654187264, 21182563584, 121263109632, 686660004864, 3851149940736, 21416533501440, 118199459288064, 647926485764096, 3529938203545600, 19124354344775680
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2024

Keywords

Crossrefs

Cf. A006139, A374497, A374511.

Programs

  • Mathematica
    a[n_]:=2^(n-4) Pochhammer[n+1, 6]*Hypergeometric2F1[(1-n)/2, -n/2, 4, 2]/45; Array[a,21,0] (* Stefano Spezia, Jul 10 2024 *)
  • PARI
    a(n) = binomial(n+6, 3)/20*sum(k=0, n\2, 2^(n-k)*binomial(n+3, n-2*k)*binomial(2*k+3, k));

Formula

a(0) = 1, a(1) = 14; a(n) = (2*(2*n+5)*a(n-1) + 4*(n+5)*a(n-2))/n.
a(n) = (binomial(n+6,3)/20) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = 2^(n-4)*Pochhammer(n+1, 6)*hypergeom([(1-n)/2, -n/2], [4], 2)/45. - Stefano Spezia, Jul 10 2024
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024

A377186 Expansion of 1/(1 - 4*x^2 - 4*x^3)^(3/2).

Original entry on oeis.org

1, 0, 6, 6, 30, 60, 170, 420, 1050, 2660, 6552, 16380, 40362, 99792, 245520, 603372, 1480050, 3624192, 8863712, 21647340, 52811616, 128700000, 313341756, 762206016, 1852565650, 4499346072, 10919990460, 26485897932, 64201490352, 155536089240, 376606931436
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2024

Keywords

Links

Crossrefs

Cf. A115962, A377189, A377190.
Cf. A374497.

Programs

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

Formula

a(0) = 1, a(1) = 0, a(2) = 6; a(n) = (4*(n+1)*a(n-2) + 2*(2*n+3)*a(n-3))/n.
a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(k,n-2*k).

A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.

Original entry on oeis.org

1, 4, 18, 80, 360, 1632, 7448, 34176, 157536, 728960, 3384128, 15754752, 73525504, 343870464, 1611288960, 7562801152, 35550504448, 167339022336, 788643765248, 3720901222400, 17573439614976, 83074892775424, 393056192851968, 1861155016212480, 8819174122700800, 41818448615636992
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2025

Keywords

Links

Crossrefs

Cf. A006139, A387402, A387403.
Cf. A071356, A374497.

Programs

  • Magma
    [&+[2^(n-k) * Binomial(n+1,n-2*k) * Binomial(2*k+1,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
  • Mathematica
    Table[Sum[2^(n-k)*Binomial[n+1,n-2*k]*Binomial[2*k+1,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+1, n-2*k)*binomial(2*k+1, k));

Formula

n*(n+2)*a(n) = (n+1) * (2*(2*n+1)*a(n-1) + 4*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+1).
E.g.f.: exp(2*x) * BesselI(1, 2*sqrt(2)*x) / sqrt(2), with offset 1.
a(n) = (n+1) * A071356(n).
Showing 1-5 of 5 results.

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