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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A178824 a(n) = Sum_{k=0..n} binomial(n,k)^4/(n+1).

Original entry on oeis.org

1, 1, 6, 41, 362, 3542, 37692, 424377, 4990722, 60704138, 758665388, 9694652838, 126203947828, 1668947978908, 22370427181624, 303383342784729, 4156846359584754, 57473870722327874, 801081711581734764, 11246487794657694810, 158920231643036635860, 2258896576436091238860
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2010

Keywords

Links

Crossrefs

Cf. A000108, A005260, A131428.

Programs

  • GAP
    List([0..20], n-> Sum([0..n], k-> Binomial(n,k)^4/(n+1) )); # G. C. Greubel, Jan 22 2019
  • Magma
    [(&+[Binomial(n,k)^4/(n+1): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jan 22 2019
  • Maple
    a:=n->add(binomial(n,k)^4/(n+1),k=0..n): seq(a(n),n=0..20); # Muniru A Asiru, Jan 22 2019
  • Mathematica
    Table[Sum[Binomial[n,k]^4/(n+1), {k,0,n}], {n,0,20}] (* G. C. Greubel, Jan 22 2019 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k)^4)/(n+1)}
  • Sage
    [sum(binomial(n,k)^4/(n+1) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jan 22 2019

Formula

a(n) = A005260(n)/(n+1).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^2. - Seiichi Manyama, Mar 26 2025
a(n) ~ 2^(4*n + 1/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Mar 26 2025

A171552 a(n) = 2^n*floor((5-2n)/3).

Original entry on oeis.org

1, 2, 0, -8, -16, -64, -192, -384, -1024, -2560, -5120, -12288, -28672, -57344, -131072, -294912, -589824, -1310720, -2883584, -5767168, -12582912, -27262976, -54525952, -117440512, -251658240, -503316480, -1073741824, -2281701376
Offset: 0

Views

Author

Paul Barry, Dec 11 2009

Keywords

Comments

Hankel transform of A131428.

Links

Crossrefs

Cf. A131428.

Formula

G.f.: (1-4x^2-16x^3)/((1-2x)(1-8x^3)).

A132808 A001263 * A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 6, 6, 1, 42, 10, 20, 10, 1, 132, 15, 50, 50, 15, 1, 429, 21, 105, 175, 105, 21, 1, 1430, 28, 196, 490, 490, 196, 28, 1, 4862, 36, 336, 1176, 1764, 1176, 336, 36, 1, 16796, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 58786, 55, 825, 4950, 13860, 19404, 13860, 4950, 825, 55, 1
Offset: 1

Views

Author

Gary W. Adamson, Aug 31 2007

Keywords

Comments

Replace left border of the Narayana triangle A001263 with the Catalan sequence A000108 starting (1, 2, 5, 14, 42, ...).
Row sums = A131428 starting (1, 3, 9, 27, 83, ...).

Examples

			First few rows of the triangle:
    1;
    2,  1
    5,  3,  1;
   14,  6,  6,  1;
   42, 10, 20, 10,  1;
  132, 15, 50, 50, 15, 1;
  ...

Crossrefs

Cf. A000108, A001263, A103451.

Extensions

Definition corrected by Philippe Deléham, Oct 11 2007
a(40) split and more terms from Georg Fischer, May 29 2023

A133200 A001263 * A136521 as infinite lower triangular matrices, where A001263 = the Narayana triangle and A136521 = an infinite lower triangular matrix with (1, 2, 2, 2, ...) in the main diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 40, 20, 2, 1, 30, 100, 100, 30, 2, 1, 42, 210, 350, 210, 42, 2, 1, 56, 392, 980, 980, 392, 56, 2, 1, 72, 672, 2352, 3528, 2352, 672, 72, 2, 1, 90, 1080, 5040, 10584, 10584, 5040, 1080, 90, 2
Offset: 1

Views

Author

Gary W. Adamson, Jan 02 2008

Keywords

Comments

Row sums = A131428 starting (1, 3, 9, 27, 83, 263, ...).

Examples

			First few rows of the triangle:
  1;
  1,  2;
  1,  6,   2;
  1, 12,  12,   2;
  1, 20,  40,  20,   2;
  1, 30, 100, 100,  30,  2;
  1, 42, 210, 350, 210, 42, 2;
  ...

Crossrefs

Cf. A001263, A136521.

A383958 Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000108(n) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 1, 7, 49, 391, 3527, 34847, 368081, 4089799, 47278087, 564211231, 6911587591, 86537984287, 1103800819999, 14305258627199, 187980039148049, 2500329655657799, 33615543148288199, 456277455475379999, 6246438365457952199, 86175353776521952799, 1197196443787879360799, 16738118900293300099199
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 16 2025

Keywords

Examples

			For n=3, the short leg is A383615(3,1) = 3 and the long leg is A383615(3,2) = 4 so the sum of the legs is then a(3) = 3 + 4 = 7.

Links

Crossrefs

Cf. A000108, A383615, A131428, A383616, A381846, A001246.

Programs

  • Mathematica
    a=Table[(2n)!/(n!(n+1)!),{n,0,23}];Apply[Join,Map[{2#^2-1}&,a]]

Formula

a(n) = A383615(n,1) + A383615(n,2).
a(n) = 2*A000108(n)^2 - 1.
a(n) = 2*A001246(n) - 1.
Previous Showing 11-15 of 15 results.

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

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