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 1-4 of 4 results.

A120305 a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * (i+j)!/(i!j!).

Original entry on oeis.org

1, 1, 3, 9, 31, 111, 407, 1513, 5679, 21471, 81643, 311895, 1196131, 4602235, 17757183, 68680169, 266200111, 1033703055, 4020716123, 15662273839, 61092127491, 238582873475, 932758045123, 3650336341239, 14298633670931
Offset: 0

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Author

Alexander Adamchuk, Jul 14 2006

Keywords

Comments

p divides a((p+1)/2) for prime p = 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, ... (A033200: primes congruent to {1, 3} mod 8; or, odd primes of the form x^2 + 2*y^2).
p divides a((p-3)/2) for prime p = 17, 41, 73, 89, 97, 113, 137, ... (A007519: primes of the form 8n+1).
Essentially the same as partial sums of A072547. - Seiichi Manyama, Jan 30 2023

Links

Crossrefs

Cf. A000108, A006134, A033200, A007519, A092785, A307354.
Cf. A072547, A371798.

Programs

  • Mathematica
    Table[Sum[Sum[(-1)^(i+j)*(i+j)!/(i!j!),{i,0,n}],{j,0,n}],{n,0,50}]
  • PARI
    a(n) = sum(i=0, n, sum(j=0, n, (-1)^(i+j) * (i+j)!/(i!*j!))); \\ Michel Marcus, Apr 02 2019
  • PARI
    a(n) = sum(i=0, 2*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^2, i)); \\ Seiichi Manyama, May 20 2019
  • PARI
    my(N=30, x='x+O('x^N)); Vec((1+sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x)))) \\ Seiichi Manyama, Jan 30 2023

Formula

a(n) = Sum_{j=0..n} Sum_{i=0..n} (-1)^(i+j)*(i+j)!/(i!j!).
Recurrence: 2*n*(3*n-5)*a(n) = 3*(9*n^2 - 19*n + 8)*a(n-1) - 3*(n-1)*(3*n-4)*a(n-2) - 2*(2*n-3)*(3*n-2)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 4^(n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 13 2013
G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( (1 - x *c(x))/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k-1,n-3*k).
a(n) = [x^n] 1/((1+x^3) * (1-x)^n). (End)

A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

1, 2, 9, 50, 295, 1792, 11088, 69498, 439791, 2803658, 17978389, 115837592, 749321716, 4863369656, 31655226108, 206549749930, 1350638103791, 8848643946550, 58069093513635, 381650672631330, 2511733593767295, 16550500379912640, 109176697072162080
Offset: 0

Views

Author

R. H. Hardin, Apr 23 2013

Keywords

Comments

Diagonal of A225010.
Number of unimodal maps [1..n]->[1..n+1], see example. - Joerg Arndt, May 10 2013

Examples

			Some solutions for n=3
..0..1..1....0..1..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..0
..1..1..1....0..1..0....1..1..1....0..0..0....0..0..0....0..1..0....0..0..1
..1..1..1....0..1..1....1..1..1....0..0..1....0..1..0....1..1..1....0..1..1
From _Joerg Arndt_, May 10 2013: (Start)
The a(2) = 9 unimodal maps [1,2]->[1,2,3] are
01:  [ 1 1 ]
02:  [ 1 2 ]
03:  [ 1 3 ]
04:  [ 2 1 ]
05:  [ 2 2 ]
06:  [ 2 3 ]
07:  [ 3 1 ]
08:  [ 3 2 ]
09:  [ 3 3 ]
(End)

Links

Crossrefs

Cf. A088536 (unimodal maps [1..n]->[1..n]).
Cf. A183160, A371798.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 0, n}]; Array[a, 30] (* Jean-François Alcover, Feb 17 2016, after Max Alekseyev *)
  • PARI
    { a(n) = polcoeff( (1+x+O(x^(2*n+1)))^(-n-1)/(1-x), 2*n) }

Formula

From Vaclav Kotesovec, May 22 2013: (Start)
Empirical: 4*n*(2*n-1)*(5*n-7)*a(n) = 2*(145*n^3 - 343*n^2 + 235*n - 48)*a(n-1) - 3*(3*n-4)*(3*n-2)*(5*n-2)*a(n-2).
a(n) ~ 3^(3*n+3/2)/(5*2^(2*n+1)*sqrt(Pi*n)). (End)
a(n) = A261668(n)+1.
a(n) = Sum_{d=0..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in (1+x)^(-n-1)/(1-x). - Max Alekseyev, Sep 14 2015
It appears that a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1+x^2) * (1-x)^(2*n)). (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 04 2017

A371836 a(n) = Sum_{k=0..floor(n/2)} n^k * binomial(2*n-2*k-1,n-2*k).

Original entry on oeis.org

1, 1, 5, 19, 91, 426, 2190, 11467, 63811, 365806, 2200978, 13677962, 88553726, 591576220, 4093814812, 29164567635, 214244414371, 1616044475734, 12523774634922, 99418836782602, 808492937082410, 6720935024074092, 57100849909374340, 495022008799053006
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2024

Keywords

Crossrefs

Cf. A293574, A371837.
Cf. A371798, A371826.

Programs

  • Mathematica
    Join[{1}, Table[Sum[n^k*Binomial[2*n-2*k-1,n-1], {k, 0, n/2}], {n, 1, 25}]] (* Vaclav Kotesovec, Apr 08 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, n^k*binomial(2*n-2*k-1, n-2*k));

Formula

a(n) = [x^n] 1/((1-n*x^2) * (1-x)^n).
a(n) ~ exp(sqrt(n) + 1/2) * n^(n/2) / 2. - Vaclav Kotesovec, Apr 08 2024

A371815 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4*n-2*k-1,n-2*k).

Original entry on oeis.org

1, 3, 20, 156, 1288, 10963, 95132, 836650, 7430956, 66501696, 598720080, 5416612336, 49201807276, 448442474938, 4099103160424, 37562606691526, 344959939645980, 3174051631201636, 29254814741949680, 270047153053464712, 2496167217049673468
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A225006, A262977, A371798.
Cf. A147855.

Programs

  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(4*n-2*k-1, n-2*k));

Formula

a(n) = [x^n] 1/((1+x^2) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, (1-n)/2, -n/2], [1/2-2*n, 1-2*n], -1). - Stefano Spezia, Apr 07 2024
Showing 1-4 of 4 results.

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