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.

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

A377189 Expansion of 1/(1 - 4*x^2 - 4*x^3)^(5/2).

Original entry on oeis.org

1, 0, 10, 10, 70, 140, 490, 1260, 3570, 9660, 25872, 69300, 182490, 480480, 1252680, 3255252, 8412690, 21655920, 55535480, 141921780, 361577216, 918529040, 2327337740, 5882631040, 14836032770, 37339221192, 93794645700, 235186913780, 588736957920, 1471462327160
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2024

Keywords

Crossrefs

Cf. A115962, A377186, A377190.
Cf. A374511.

Programs

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

Formula

a(0) = 1, a(1) = 0, a(2) = 10; a(n) = (4*(n+3)*a(n-2) + 2*(2*n+9)*a(n-3))/n.
a(n) = Sum_{k=0..floor(n/2)} (-4)^k * binomial(-5/2,k) * binomial(k,n-2*k).

A377190 Expansion of 1/(1 - 4*x^2 - 4*x^3)^(7/2).

Original entry on oeis.org

1, 0, 14, 14, 126, 252, 1050, 2772, 8778, 24948, 72072, 204204, 570570, 1585584, 4351776, 11879868, 32162130, 86582496, 231703472, 616900284, 1634721088, 4312944064, 11333823228, 29673291648, 77423101938, 201367680696, 522180220044, 1350350044316, 3482928560880
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2024

Keywords

Crossrefs

Cf. A115962, A377186, A377189.
Cf. A374513.

Programs

  • PARI
    a(n) = sum(k=0, n\2, (-4)^k*binomial(-7/2, k)*binomial(k, n-2*k));

Formula

a(0) = 1, a(1) = 0, a(2) = 14; a(n) = (4*(n+5)*a(n-2) + 2*(2*n+15)*a(n-3))/n.
a(n) = Sum_{k=0..floor(n/2)} (-4)^k * binomial(-7/2,k) * binomial(k,n-2*k).
Showing 1-3 of 3 results.

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

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