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.

A298567 a(n) = Sum_{k=0..2*n/3} C(n-k,2*k-n)^2.

Original entry on oeis.org

1, 0, 1, 1, 1, 4, 2, 9, 10, 17, 37, 41, 102, 136, 251, 450, 667, 1325, 2011, 3658, 6246, 10293, 18686, 30461, 54183, 92169, 157438, 276414, 466579, 818256, 1400509, 2419379, 4202829, 7208342, 12556360, 21621891, 37480728, 64965461, 112227269
Offset: 0

Views

Author

Vladimir Kruchinin, Jan 21 2018

Keywords

Crossrefs

Cf. A182878.

Programs

  • Maple
    A298567 := proc(n)
    option remember;
    if n < 7 then
        op(n+1,[1, 0, 1, 1, 1, 4,2]) ;
    else
        -2*(n-1)*procname(n-2)-(2*n-3)*procname(n-3)+(n-2)*procname(n-4)
            -(2*n-5)*procname(n-5)+(n-3)*procname(n-6) ;
        -%/n ;
    end if;
end proc: # R. J. Mathar, Jan 21 2020
  • Maxima
    a(n):=sum(binomial(n-k,2*k-n)^2,k,0,2*n/3);

Formula

G.f.: 1/sqrt((1-x^2)^2+x^6-2*x^5-2*x^3).
D-finite with recurrence: n*a(n) -2*(n-1)*a(n-2)-(2*n-3)*a(n-3)+(n-2)*a(n-4) -(2*n-5)*a(n-5) +(n-3)*a(n-6) = 0. - R. J. Mathar, Jan 21 2020

A182879 The sum of the lengths of all weighted lattice paths in L_n.

Original entry on oeis.org

0, 1, 3, 11, 33, 96, 278, 787, 2205, 6133, 16941, 46554, 127390, 347331, 944121, 2559607, 6923529, 18690138, 50364988, 135506485, 364063815, 976880631, 2618206923, 7009868646, 18749876418, 50107633501, 133800148323, 357012426677, 951936494055
Offset: 0

Views

Author

Emeric Deutsch, Dec 11 2010

Keywords

Comments

L_n is the set of lattice paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Examples

			a(3)=11. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), there are five paths of weight 3: hhh, hH, Hh, ud, and du; their lengths are 3, 2, 2, 2,and 2, respectively.

Links

Crossrefs

Cf. A182878.

Programs

  • Maple
    a:=n->sum(k*binomial(k,n-k)^2,k=0..n): seq(a(n),n=0..28);
  • Mathematica
    CoefficientList[Series[x*(1+2*x^2-x^3)/((1-3*x+x^2)*(1+x+x^2))^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)
  • PARI
    z='z+O('z^50); concat([0], Vec(z*(1+2*z^2-z^3)/((1-3*z+z^2)*(1+z+z^2))^(3/2))) \\ G. C. Greubel, Mar 25 2017

Formula

a(n) = Sum_{k=0..n} k*binomial(k,n-k)^2.
G.f.: z*(1+2*z^2-z^3)/((1-3*z+z^2)*(1+z+z^2))^(3/2).
a(n) ~ sqrt(20 + 9*sqrt(5)) * ((3 + sqrt(5))/2)^n * sqrt(n) / (10*sqrt(Pi)). - Vaclav Kotesovec, Mar 06 2016
Conjecture: (n-1)*(310*n-781)*a(n) + (-882*n^2+3845*n-3679)*a(n-1) + (214*n^2-2903*n+3751)*a(n-2) + (-358*n^2-459*n+1955)*a(n-3) + (834*n^2-3631*n+4065)*a(n-4) - (262*n-663)*(n-3)*a(n-5) = 0. - R. J. Mathar, Jun 14 2016
Recurrence (of order 4): (n-1)*(2*n-5)*(2*n^2 - 15*n + 24)*a(n) = 2*(n-3)*(4*n^3 - 30*n^2 + 52*n - 13)*a(n-1) + (4*n^4 - 40*n^3 + 133*n^2 - 201*n + 96)*a(n-2) + 2*(4*n^4 - 38*n^3 + 110*n^2 - 126*n + 51)*a(n-3) - (n-2)*(2*n-3)*(2*n^2 - 11*n + 11)*a(n-4). - Vaclav Kotesovec, Sep 23 2017
a(n) ~ phi^(2*n + 3) * sqrt(n) / (2 * 5^(3/4) * sqrt(Pi)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017

A205457 Symmetric matrix, by antidiagonals: C(max(2i,2j),min(2i,2j)), i>=0, j>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 15, 1, 15, 1, 1, 28, 15, 15, 28, 1, 1, 45, 70, 1, 70, 45, 1, 1, 66, 210, 28, 28, 210, 66, 1, 1, 91, 495, 210, 1, 210, 495, 91, 1, 1, 120, 1001, 924, 45, 45, 924, 1001, 120, 1, 1, 153, 1820, 3003, 495, 1, 495, 3003, 1820, 153, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1....6....15...28...45
6....1....15...70...210
15...15...1....28...210
28...70...28...1....45
45...210..210..45...1

Crossrefs

Cf. A182878, A205456.

Programs

  • Mathematica
    f[i_, j_] := Binomial[Max[2 i - 2, 2 j - 2], Min[2 i - 2, 2 j - 2]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]
  • Maxima
    S(x,y):=((x^3-3*x^2)*y^3-x^2*y^2)/((x^2-2*x+1)*y^3+(-x^2-3)*y^2+(2*x+3)*y-1);
taylor((S(x,y)+S(y,x))/(x*y)-x*y/(1-x*y)+1/(1-x)+1/(1-y)-1,x,0,7,y,0,7); /* Vladimir Kruchinin, Oct 29 2020 */

Formula

G.f.: (S(x,y)+S(y,x))/(x*y)-x*y/(1-x*y)+1/(1-x)+1/(1-y)-1, where S(x,y)=((x^3-3*x^2)*y^3-x^2*y^2)/((x^2-2*x+1)*y^3+(-x^2-3)*y^2+(2*x+3)*y-1). - Vladimir Kruchinin, Oct 29 2020

A370232 Triangle read by rows. T(n, k) = binomial(n + k, 2*k)^2.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 36, 25, 1, 1, 100, 225, 49, 1, 1, 225, 1225, 784, 81, 1, 1, 441, 4900, 7056, 2025, 121, 1, 1, 784, 15876, 44100, 27225, 4356, 169, 1, 1, 1296, 44100, 213444, 245025, 81796, 8281, 225, 1, 1, 2025, 108900, 853776, 1656369, 1002001, 207025, 14400, 289, 1
Offset: 0

Views

Author

Peter Luschny, Feb 12 2024

Keywords

Examples

			Triangle starts:
[0] 1;
[1] 1,     1;
[2] 1,     9,       1;
[3] 1,    36,      25,       1;
[4] 1,   100,     225,      49,       1;
[5] 1,   225,    1225,     784,      81,      1;
[6] 1,   441,    4900,    7056,    2025,    121,     1;
[7] 1,   784,   15876,   44100,   27225,   4356,   169,   1;

Crossrefs

Shifted bisection of A182878.
Cf. A370233 (c=2), A188648 (row sums), A188662 (central terms).

Programs

  • Mathematica
    Table[Binomial[n + k, 2*k]^2, {n, 0, 7}, {k, 0, n}] // Flatten

Formula

T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n + k, 2*k) * Pochhammer(n - k + c, 2*k) * z^k / (2*k)! and c = 1.
T(n, k) = [z^k] hypergeom([-n, -n, 1 + n, 1 + n], [1/2, 1/2, 1], z/16).
Showing 1-4 of 4 results.

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