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 1-5 of 5 results.

A098506 Row sums of the number triangle A098505.

Original entry on oeis.org

1, 2, 3, 8, 9, 22, 19, 100, 101, 266, 435, 860, 1603, 4110, 6529, 4672, 12397, 21558, 7845, 57648, 9447, 50190, 199053, 1880620, 1710309, 7344462, 7529113, 34610408, 555889, 3316906, 21528385, 167087336, 124402817, 73289470, 764401867
Offset: 0

Views

Author

Paul Barry, Sep 11 2004

Keywords

Links

Programs

  • Mathematica
    Table[Sum[Numerator[((n+1)Binomial[n,k])/Binomial[2n,n]],{k,0,n}],{n,0,40}] (* Harvey P. Dale, Nov 18 2012 *)

Formula

a(n)=sum{k=0..n, numerator((n+1)binomial(n, k)/binomial(2n, n))}

A098508 Second column of the inverse of a Catalan scaled binomial matrix (A098505).

Original entry on oeis.org

0, 0, 1, 3, 3, 5, 5, 7, 14, 18, 45, 55, 33, 39, 91, 7, 4, 68, 51, 57, 19, 7, 77, 253, 23, 25, 325, 351, 27, 29, 435, 465, 248, 88, 187, 85, 45, 333, 703, 741, 39, 41, 41, 43, 473, 11, 23, 1081, 94, 98, 1225, 425, 221, 689, 477, 495, 385, 133, 551, 1711, 295, 305, 1891
Offset: 0

Views

Author

Paul Barry, Sep 11 2004

Keywords

Links

Crossrefs

Second column of A098505.

Programs

  • Mathematica
    Table[Numerator[n*(n^2 - 1)/(2*Binomial[2*n, n])], {n, 0, 100}] (* Paolo Xausa, Aug 30 2024 *)

Formula

a(n) = numerator((n+1)*binomial(n, 2)/binomial(2*n, n));
a(n) = numerator((n-1)*n*(n+1)/(2*binomial(2*n, n))).

A098507 Diagonal sums of the number triangle A098505.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 10, 9, 19, 22, 65, 88, 114, 187, 395, 376, 888, 520, 2645, 2017, 3303, 3590, 8820, 9408, 5785, 7792, 24181, 27047, 74720, 68731, 153394, 159394, 369957, 1261522, 4326030, 4924898, 6538595, 9754155, 10068484, 10541264, 24936678
Offset: 0

Views

Author

Paul Barry, Sep 11 2004

Keywords

Crossrefs

Cf. A098505.

Programs

  • Mathematica
    Table[Sum[Numerator[(n-k+1) Binomial[n-k,k]/Binomial[2n-2k,n-k]],{k,0,Floor[ n/2]}],{n,0,60}] (* Harvey P. Dale, Jun 19 2021 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} numerator((n-k+1)*binomial(n-k, k)/binomial(2*n-2*k, n-k)).

A093527 Denominators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

Original entry on oeis.org

1, 1, 3, 2, 5, 1, 7, 4, 9, 5, 11, 3, 13, 7, 1, 8, 17, 3, 19, 1, 7, 11, 23, 2, 25, 13, 27, 1, 29, 15, 31, 16, 11, 17, 5, 9, 37, 19, 39, 2, 41, 1, 43, 11, 1, 23, 47, 4, 49, 25, 17, 13, 53, 9, 55, 7, 19, 29, 59, 5, 61, 31, 21, 32, 13, 1, 67, 17, 23, 7, 71, 2, 73, 37, 5, 19, 1, 13, 79
Offset: 0

Views

Author

Eric W. Weisstein, Mar 30 2004

Keywords

Examples

			1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...

Links

Crossrefs

Cf. A093070, A093526, A093528, A093529.
Second column of A098505.
Cf. A000108.

Programs

Formula

a(k) = Denominator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k.
From Paul Barry, Sep 11 2004: (Start)
a(n) = numerator((n+1)(n+2)/binomial(2(n+1), n+1));
a(n) = numerator(2*binomial(n+2, 2)/binomial(2(n+1), n+1)). (End)
a(n) = numerator((n+1)/C(n+1)). - Paul Barry, Nov 17 2004
a(n) = denominator(binomial(2n, n)/n). - Enrique Pérez Herrero, Oct 05 2011
a(n) = n/gcd(n,binomial(2n,n)). - Peter Luschny, Oct 05 2011
a(n) = denominator((n + 2)*binomial(2*n+3, n+1)/((n + 1)*(2*n + 3))). - Stefano Spezia, Aug 06 2022

A098509 Denominators of the inverse of a Catalan scaled binomial matrix.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 5, 5, 5, 14, 7, 7, 7, 14, 42, 42, 21, 21, 42, 42, 132, 22, 44, 33, 44, 22, 132, 429, 429, 143, 429, 429, 143, 429, 429, 1430, 715, 715, 715, 143, 715, 715, 715, 1430, 4862, 4862, 2431, 2431, 2431, 2431, 2431, 2431, 4862, 4862, 16796, 8398
Offset: 0

Views

Author

Paul Barry, Sep 11 2004

Keywords

Comments

First column and main diagonal are A000108. Row sums are A098510. Diagonal sums are A098511. Second column is A098512.

Examples

			Rows begin
1;
1,1;
2,1,2;
5,5,5,5;
14,7,7,7,14;
...

Crossrefs

Cf. A098505, A098474.

Programs

  • Mathematica
    T[n_, k_] := Denominator[(n + 1)*(Binomial[n, k]/Binomial[2*n, n])];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2018 *)

Formula

Triangle T(n, k)=denominator((n+1)binomial(n, k)/binomial(2n, n))
Showing 1-5 of 5 results.

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

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