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.

A243667 Number of Sylvester classes of 4-packed words of degree n.

Original entry on oeis.org

1, 1, 6, 50, 484, 5105, 56928, 660112, 7878940, 96159476, 1194532794, 15053992178, 191993403476, 2473358617150, 32137897641232, 420698195672700, 5542894551818268, 73447821835338348, 978178443083177880, 13086377223959022952, 175785879063917657688
Offset: 0

Views

Author

N. J. A. Sloane, Jun 14 2014

Keywords

Comments

See Novelli-Thibon (2014) for precise definition.

Links

Crossrefs

Column k=4 of A336573.
Cf. A243668, A336572.

Programs

  • Mathematica
    P[n_, m_, x_] := 1/(m n + 1) Sum[Binomial[m n + 1, k] Binomial[(m + 1) n - k, n - k] (1 - x)^k x^(n - k), {k, 0, n}];
a[n_] := P[n, 4, 2];
a /@ Range[20] (* Jean-François Alcover, Jan 28 2020 *)
  • PARI
    a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^4*(1-2*A)); polcoeff(A, n); \\ Seiichi Manyama, Jul 26 2020
  • PARI
    a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1)); \\ Seiichi Manyama, Jul 26 2020
  • PARI
    a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(4*n+1, k)*binomial(5*n-k, n-k))/(4*n+1); \\ Seiichi Manyama, Jul 26 2020

Formula

Novelli-Thibon give an explicit formula in Eq. (182).
From Seiichi Manyama, Jul 26 2020: (Start)
G.f. A(x) satisfies: A(x) = 1 - x * A(x)^4 * (1 - 2 * A(x)).
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).
a(n) = ( (-1)^n / (4*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k) * binomial(5*n-k,n-k). (End)
a(n) ~ 2^(9*n - 15) * sqrt(436289 + 2793997/sqrt(41)) / (sqrt(Pi) * n^(3/2) * (29701 - 4633*sqrt(41))^(n - 1/2)). - Vaclav Kotesovec, Jul 31 2021
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023

Extensions

More terms from Jean-François Alcover, Jan 28 2020
a(0)=1 prepended by Seiichi Manyama, Jul 25 2020

A336573 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (-1)^n * Sum_{j=0..n} (-2)^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 8, 1, 1, 5, 21, 45, 16, 1, 1, 6, 34, 126, 197, 32, 1, 1, 7, 50, 267, 818, 903, 64, 1, 1, 8, 69, 484, 2279, 5594, 4279, 128, 1, 1, 9, 91, 793, 5105, 20540, 39693, 20793, 256, 1, 1, 10, 116, 1210, 9946, 56928, 192350, 289510, 103049, 512
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2020

Keywords

Comments

T(n,k) is the number of Sylvester classes of k-packed words of degree n.

Examples

			Square array begins:
   1,   1,   1,    1,    1,    1, ...
   1,   1,   1,    1,    1,    1, ...
   2,   3,   4,    5,    6,    7, ...
   4,  11,  21,   34,   50,   69, ...
   8,  45, 126,  267,  484,  793, ...
  16, 197, 818, 2279, 5105, 9946, ...

Links

Crossrefs

Columns k = 0-5 are: A011782, A001003, A003168, A243659, A243667, A243668.
Main diagonal is A336495.
Cf. A336534, A336574, A336575.

Programs

  • Maple
    T := (n,k) -> `if`(k=0, `if`(n=0, 1, 2^(n-1)), (-1)^n*(binomial(k*n+1, n)* hypergeom([-n, k*n+1], [(k-1)*n+2], 2)) / (k*n+1)):
seq(lprint(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, Jul 26 2020
  • Mathematica
    T[n_, k_] := (-1)^n * Sum[(-2)^j * Binomial[n, j] * Binomial[k*n+j+1, n]/(k*n+j+1), {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
  • PARI
    T(n, k) = (-1)^n*sum(j=0, n, (-2)^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));
  • PARI
    T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^k*(1-2*A)); polcoeff(A, n);
  • PARI
    T(n, k) = (-1)^n*sum(j=0, n, (-2)^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 - x * A_k(x)^k * (1 - 2 * A_k(x)).
T(n,k) = ( (-1)^n / (k*n+1) ) * Sum_{j=0..n} (-2)^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
T(n,k) = (-1)^n*binomial(k*n+1, n)*hypergeom([-n, k*n+1], [(k-1)*n+2], 2)/(k*n+1) for k >= 1. - Peter Luschny, Jul 26 2020
T(n,k) = (1/n) * Sum_{j=0..n-1} binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0. - Seiichi Manyama, Aug 08 2023

A364827 G.f. satisfies A(x) = 1 - x*A(x)^5 * (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 26, 478, 10254, 240122, 5950530, 153417542, 4072868742, 110585691634, 3056671795946, 85722961493742, 2433127206219582, 69763483031049066, 2017643094336224914, 58789801741123032918, 1724199860717303739062, 50858327392484088101346
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2023

Keywords

Links

Crossrefs

Cf. A025192, A107841, A235347, A364825, A364826.
Cf. A243668, A363006.

Programs

  • PARI
    a(n) = (-1)^n*sum(k=0, n, (-3)^k*binomial(n, k)*binomial(5*n+k+1, n)/(5*n+k+1));

Formula

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(5*n+k+1,n) / (5*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(5*n,k-1) for n > 0.
Showing 1-3 of 3 results.

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

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