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.

A013561 Erroneous version of A348211.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 31, 90, 31, 1, 1, 85, 544, 544, 85, 1, 1, 225, 2997, 6559, 2997, 225, 1
Offset: 3

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Author

Keywords

Links

A012249 Volume of a certain rational polytope whose points with given denominator count certain sets of Standard Tableaux.

Original entry on oeis.org

1, 2, 5, 24, 154, 1280, 13005, 156800, 2189726, 34793472, 620169186, 12259602432, 266267950740, 6304157663232, 161624247752253, 4461403146190848, 131936409635518774, 4161949856324648960, 139508340802911502422, 4952126960969786064896, 185585825504872433198636
Offset: 1

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Author

D n Verma

Keywords

Comments

It should be noticed that Richard Stanley's formula (cf. A012250) gives a(9) = 2189726 instead of 2189725 as given in Verma (1997). - Jean-François Alcover, Nov 28 2013

Links

Crossrefs

Cf. A012250.
Row sums of A348211.

Programs

  • Magma
    A012249:= func< n | 2^(n-2)*(&+[(-1)^(j+1)*Binomial(n+2,j)*(n/2-j+1)^(n-1) : j in [0..1+Floor(n/2)]] ) >;
[A012249(n): n in [1..30]]; // G. C. Greubel, Feb 28 2024
  • Maple
    A012249 := proc(n)
     add( (-1)^(j+1)*(n/2-j+1)^(n-1)*binomial(n+2,j),j=0..ceil(n/2)) ;
     %*2^(n-2) ;
end proc:
seq(A012249(n),n=1..20) ; # R. J. Mathar, Oct 07 2021
  • Mathematica
    a[n_] := 2^(n-2)*Sum[(-1)^(j+1)*(n/2-j+1)^(n-1)*Binomial[n+2, j], {j, 0, Ceiling[n/2]}]; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Nov 25 2013, after Richard Stanley's formula in A012250. *)
  • SageMath
    def A012249(n): return 2^(n-2)*sum( (-1)^(j+1)*binomial(n+2,j)*(n/2-j+1)^(n-1) for j in range(n//2+2))
[A012249(n) for n in range(1,31)] # G. C. Greubel, Feb 28 2024

Formula

a(n) ~ 3^(3/2) * 2^(n+1) * n^(n-2) / exp(n). - Vaclav Kotesovec, Oct 07 2021
a(n) = 2^(n-2)*Sum_{j=0..ceiling(n/2)} (-1)^(j+1)*(n/2-j+1)^(n-1) * binomial(n+2, j) (based on Richard Stanley's formula in A012250). - Jean-François Alcover, Nov 25 2013

Extensions

Corrected and extended by R. J. Mathar, Oct 07 2021
Edited by N. J. A. Sloane, Oct 07 2021

A348210 Varma's Kosta numbers of semi-standard tableaux: array A(n>=2, k>=0) read by rising antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 15, 16, 7, 1, 0, 1, 36, 65, 31, 9, 1, 0, 1, 91, 260, 175, 51, 11, 1, 0, 1, 232, 1085, 981, 369, 76, 13, 1, 0, 1, 603, 4600, 5719, 2661, 671, 106, 15, 1, 0, 1, 1585, 19845, 33922, 19929, 5916, 1105, 141, 17, 1, 0, 1, 4213, 86725, 204687, 151936, 54131, 11516, 1695, 181, 19, 1, 0
Offset: 2

Views

Author

R. J. Mathar, Oct 07 2021

Keywords

Comments

(More characteristic NAME desired.)
Each row is a polynomial in k, which implies that the inverse binomial transformation of each row is a finite sequence and that the row can be represented by a rational generating function (A348211).

Examples

			The array starts in row n=2 with columns k>=0 as:
  0   0    0    0     0     0      0      0 ...
  1   1    1    1     1     1      1      1 ...
  1   3    5    7     9    11     13     15 ...
  1   6   16   31    51    76    106    141 ...
  1  15   65  175   369   671   1105   1695 ...
  1  36  260  981  2661  5916  11516  20385 ...
  1  91 1085 5719 19929 54131 124501 254255 ...
Antidiagonal rows begin as:
  0;
  1,   0;
  1,   1,    0;
  1,   3,    1,    0;
  1,   6,    5,    1,    0;
  1,  15,   16,    7,    1,    0;
  1,  36,   65,   31,    9,    1,   0;
  1,  91,  260,  175,   51,   11,   1,   0;
  1, 232, 1085,  981,  369,   76,  13,   1,  0;
  1, 603, 4600, 5719, 2661,  671, 106,  15,  1,  0;

Links

Crossrefs

Cf. A005043 (column k=1), A007043 (k=2), A264608 (k=3), A272393 (k=4), A005408 (row n=4), A005891 (n=5), A005917 (n=6), A348211 (condensed g.f.)

Programs

  • Magma
    A:= func< n,k | (&+[(-1)^(j+1)*Binomial(n,j)*Binomial((n-2*j)*k+n-j-2,n-3)/2 : j in [0..Floor((n-1)/2)]]) >;
A348210:= func< n,k | A(n-k,k) >;
[A348210(n,k): k in [0..n-2], n in [2..13]]; // G. C. Greubel, Feb 28 2024
  • Maple
    A348210 := proc(n,k)
    local a,j ;
    a := 0 ;
    for j from 0 to floor((n-1)/2) do
            a := a+ (-1)^j *binomial(n,j) *binomial( (n-2*j)*k+n-j-2,n-3) ;
    end do:
    -a/2 ;
end proc:
seq( seq( A348210(d-k,k),k=0..d-2),d=2..12) ;
  • Mathematica
    A[n_, k_] := (-1/2)*Sum[(-1)^j*Binomial[n, j]*Binomial[(n - 2*j)*k + n - j - 2, n - 3], {j, 0, Floor[(n - 1)/2]}];
Table[A[n - k, k], {n, 2, 13}, {k, 0, n - 2}] // Flatten (* Jean-François Alcover, Mar 06 2023 *)
  • SageMath
    def A(n,k): return sum( (-1)^(j+1)*binomial(n,j)*binomial((n-2*j)*k+n-j-2,n-3) for j in range(1+(n-1)//2) )/2
def A348210(n,k): return A(n-k, k)
flatten([[A348210(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, Feb 28 2024

Formula

A(n,k) = (-1/2)*Sum_{j=0..floor((n-1)/2)} (-1)^j *binomial(n,j) *binomial((n-2*j)*k+n-j-2,n-3).
A(7,k) = 1 + 7*k*(k+1)*(11*k^2+11*k+8)/12.
A(8,k) = (2*k+1)*(4*k^2+6*k+3)*(4*k^2+2*k+1)/3.
A(9,k) = 1 + k*(k+1)*(289*k^4+578*k^3+581*k^2+292*k+108)/16.
Showing 1-3 of 3 results.

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