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.

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A378816 Expansion of 2*(x - 1)^3/(3*x^3 - 5*x^2 + x + 1 + sqrt(-(x - 1)^3*(x + 1)^2*(3*x + 1))).

Original entry on oeis.org

-1, 4, -11, 30, -83, 232, -654, 1856, -5296, 15180, -43675, 126062, -364863, 1058552, -3077533, 8963862, -26151753, 76409052, -223544241, 654790218, -1920055017, 5635816776, -16557539124, 48685404516, -143264248974, 421879104836, -1243160223829, 3665516301186
Offset: 0

Views

Author

Thomas Scheuerle, Dec 08 2024

Keywords

Comments

Binomial transform of A057552(n)*(-1)^(n+1).

Crossrefs

Cf. A025566, A057552, A378783, A378816 ( Hankel sequence transform ).

Programs

  • PARI
    a(n) = sum(k=1, n+1, binomial(n, k-1)*(-1)^k*sum(m=0, k-1, binomial(2*m+2, m)))

Formula

G.f. A(x) satisfies: (-3*x^3 - x^2)*A(x)^2 + (3*x^3 - 5*x^2 + x + 1)*A(x) + (-x^3 + x*y^2 - x*y + 1) = 0.
a(n) = Limit_{k->oo} (A378783(k, k-n) - A378783(k, k-n-1)).
a(n) = A025566(n+1)+A025566(n+2)*(-1)^(n+1), for n > 0.
a(n) = Sum_{k=1..n+1} binomial(n, k-1)*(-1)^k*Sum_{m=0..k-1} binomial(2*m+2, m).

A378817 Hankel sequence transform of A378816.

Original entry on oeis.org

-1, -5, 6, -10, 11, -1, 1, 17, -18, 22, -23, 1, -1, -29, 30, -34, 35, -1, 1, 41, -42, 46, -47, 1, -1, -53, 54, -58, 59, -1, 1, 65, -66, 70, -71, 1, -1, -77, 78, -82, 83, -1, 1, 89, -90, 94, -95, 1, -1, -101, 102, -106, 107, -1, 1, 113, -114, 118, -119
Offset: 0

Views

Author

Thomas Scheuerle, Dec 08 2024

Keywords

Links

Crossrefs

Cf. A378783, A378816.

Programs

  • Mathematica
    LinearRecurrence[{0, 2, 0, -3, 0, 2, 0, -1}, {-1, -5, 6, -10, 11, -1, 1, 17}, 100] (* Paolo Xausa, Jan 28 2025 *)
  • PARI
    a(n) =  (1+(2*n+3-n%3)*((n%6)*((n+1)%6)>0))*(-1)^(n+1+ceil(n/6))

Formula

G.f.: (-1 - 5*x + 8*x^2 - 4*x^4 + 4*x^5 - x^6 - x^7)/(1 - x^2 + x^4)^2.
a(n) = 2*a(n-2) - 3*a(n-4) + 2*a(n-6) - a(n-8).
a(12*n) = -1.
a(5+12*n) = -1.
a(6+12*n) = 1.
a(11+12*n) = 1.
a(1+12*n) = -5-24*n.
a(2+12*n) = 6+24*n.
a(3+12*n) = -10-24*n.
a(4+12*n) = 11+24*n.
a(7+12*n) = 17+24*n.
a(8+12*n) = -18-24*n.
a(9+12*n) = 22+24*n.
a(10+12*n) = -23-24*n.
Showing 1-2 of 2 results.

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

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