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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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

A336574 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 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, 3, 1, 3, 6, 1, 3, 15, 12, 1, 3, 24, 93, 24, 1, 3, 33, 255, 645, 48, 1, 3, 42, 498, 3102, 4791, 96, 1, 3, 51, 822, 8691, 40854, 37275, 192, 1, 3, 60, 1227, 18708, 164937, 566934, 299865, 384, 1, 3, 69, 1713, 34449, 464115, 3305868, 8164263, 2474025, 768
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2020

Keywords

Examples

			Square array begins:
   1,    1,     1,      1,      1,       1, ...
   3,    3,     3,      3,      3,       3, ...
   6,   15,    24,     33,     42,      51, ...
  12,   93,   255,    498,    822,    1227, ...
  24,  645,  3102,   8691,  18708,   34449, ...
  48, 4791, 40854, 164937, 464115, 1055838, ...

Links

Crossrefs

Columns k=0-4 give: A003945, A103210, A219536, A336539, A336572.
Main diagonal gives A336577.
Cf. A336534, A336573, A336575.

Programs

  • Mathematica
    T[n_, k_] := 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, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 27 2020 *)
  • PARI
    T(n, k) = 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) = 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/(k*n+1)) * Sum_{j=0..n} 2^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
From Seiichi Manyama, Aug 10 2023: (Start)
T(n,k) = (1/n) * Sum_{j=0..n-1} (-1)^j * 3^(n-j) * binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0.
T(n,k) = (1/n) * Sum_{j=1..n} 3^j * 2^(n-j) * binomial(n,j) * binomial(k*n,j-1) for n > 0. (End)
T(n,k) = binomial(1+k*n, n)*hypergeom([-n, 1+k*n], [2+(k-1)*n], -2)/(1 + k*n) for k > 0. - Stefano Spezia, Aug 09 2025
Showing 1-2 of 2 results.

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

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