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 1-7 of 7 results.

A108424 Number of paths from (0,0) to (3n,0) that stay in the first quadrant, consist of steps u=(2,1), U=(1,2), or d=(1,-1) and do not touch the x-axis, except at the endpoints.

Original entry on oeis.org

2, 6, 34, 238, 1858, 15510, 135490, 1223134, 11320066, 106830502, 1024144482, 9945711566, 97634828354, 967298498358, 9659274283650, 97119829841854, 982391779220482, 9990160542904134, 102074758837531810, 1047391288012377774, 10788532748880319298
Offset: 1

Views

Author

Emeric Deutsch, Jun 03 2005

Keywords

Comments

These are the large nu-Schröder numbers with nu=NE(NEE)^(n-1). - Matias von Bell, Jun 02 2021

Examples

			a(2) = 6 because we have uudd, uUddd, Ududd, UdUddd, Uuddd and UUdddd.

Links

Crossrefs

Cf. A027307, A032349, A344553.
Cf. A006318 (d = 2, signed version at d = 0), A027307 (d = 3), A144097 (d = 4), A260332 (d = 5, conjecturally), A363006 (d = 6).

Programs

  • Maple
    A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=z*A+z*A^2: Gser:=series(G,z=0,28): seq(coeff(Gser,z^n),n=1..25);
a:=proc(n) if n=1 then 2 else (n*2^n*binomial(2*n,n)/((2*n-1)*(n+1)))*sum(binomial(n-1,j)^2/2^j/binomial(n+j+1,j),j=0..n-1) fi end: seq(a(n),n=1..19);
# Alternative:
a := n -> 2*binomial(3*n - 2, 2*n - 1)*hypergeom([2 - 2*n, 1 - n], [2 - 3*n], -1)/n:
seq(simplify(a(n)), n = 1..21); # Peter Luschny, Jun 14 2021
  • Mathematica
    Table[(n*2^n*Binomial[2*n,n]/((2n-1)*(n+1))) * Sum[(Binomial[n-1,j])^2/ (2^j * Binomial[n+j+1,j]), {j,0,n-1}], {n,1,20}] (* Vaclav Kotesovec, Oct 17 2012 *)

Formula

a(n) = A027307(n-1) + A032349(n).
G.f.: z*A+z*A^2, where A=1+z*A^2+z*A^3 or, equivalently, A=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3.
a(n) = (n*2^n*C(2*n, n)/((2n-1)(n+1))) * Sum_{j=0..n-1} (C(n-1, j))^2 / (2^j*C(n+j+1,j)).
Recurrence: n*(2*n-1)*a(n) = 3*(6*n^2-10*n+3)*a(n-1) + (46*n^2-227*n+279)*a(n-2) + 2*(n-3)*(2*n-7)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(30*sqrt(5) - 50)*((11 + 5*sqrt(5))/2)^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n) = Sum_{i=0..n} (2*n+i-2)!/((n-i)!*(n+i-1)!*i!), n>0. - Vladimir Kruchinin, Feb 16 2013
From Matias von Bell, Jun 02 2021: (Start)
a(n) = 2*Sum_{i>=0} (1/n)*binomial(2*n-2,i)*binomial(3*n-2-i,2*n-1).
a(n) = 2*A344553(n). (End)
a(n) = 2*binomial(3*n - 2, 2*n - 1)*hypergeom([2 - 2*n, 1 - n], [2 - 3*n], -1) / n. - Peter Luschny, Jun 14 2021
From Peter Bala, Jun 17 2023: (Start)
a(n) = (-1)^(n+1) * (1/((d-1)*n + 1))*Sum_{i = 0..n} binomial((d - 1)*n+1, n-i) * binomial((d-1)*n+i, i), with d = -1.
P-recursive: n*(2*n - 1)*(5*n - 8)*a(n) = (110*n^3 - 396*n^2 + 445*n - 150)*a(n-1) + (n - 2)*(2*n - 5)*(5*n - 3)*a(n-2) with a(1) = 2 and a(2) = 6.
The g.f. A(x) = 2*x + 6*x^2 + 34*x^3 + .... Then 1/(1 - A(x)) = 1 + 2*x + 10*x^2 + 66*x^3 + .. is the g.f. of A027307.
(1/x) * the series reversion of x*(1 - A(x)) = 1 + 2*x + 14*x^2 + 134*x^3 + ... is the g.f. of A144097.
(1/x) * the series reversion of x/(1 - A(x)) = 1 - 2*x - 2*x^2 - 6*x^3 - 22*x^4 - 90*x^5 - ... = 1 - x - x*S(x), where S(x) is the g.f. of A006318. (End)

A364827 G.f. satisfies A(x) = 1 - x*A(x)^5 * (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 26, 478, 10254, 240122, 5950530, 153417542, 4072868742, 110585691634, 3056671795946, 85722961493742, 2433127206219582, 69763483031049066, 2017643094336224914, 58789801741123032918, 1724199860717303739062, 50858327392484088101346
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2023

Keywords

Links

Crossrefs

Cf. A025192, A107841, A235347, A364825, A364826.
Cf. A243668, A363006.

Programs

  • PARI
    a(n) = (-1)^n*sum(k=0, n, (-3)^k*binomial(n, k)*binomial(5*n+k+1, n)/(5*n+k+1));

Formula

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(5*n+k+1,n) / (5*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(5*n,k-1) for n > 0.

A371677 G.f. satisfies A(x) = 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 48, 772, 14256, 285380, 6023552, 131991940, 2974096544, 68475379204, 1603913377040, 38099316926340, 915619571011024, 22222175033464260, 543894269096547296, 13409307961403740420, 332707806061304185408, 8301493488646359256580
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Crossrefs

Cf. A006319, A032349, A371675. A371676, A371678.
Cf. A363006.

Programs

  • PARI
    a(n, r=2, t=5, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

G.f. satisfies A(x) = ( 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A363006.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(5*n+k+2,n)/(5*n+k+2).

A364195 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^5 * (1 + A(x)^2).

Original entry on oeis.org

1, 2, 24, 412, 8280, 181904, 4232048, 102479184, 2555884896, 65207430848, 1693785940992, 44643489969792, 1190986788639232, 32097745138518528, 872595854798515456, 23900545715576753408, 658934625866433496576, 18271554709525993556992, 509241947434834351042560
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2023

Keywords

Crossrefs

Cf. A217364, A363006, A363305, A364196.
Cf. A349310, A363304, A363311.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+2*k+1, n)/(5*n+2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+2*k+1,n)/(5*n+2*k+1).

A364196 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^5 * (1 + A(x)^3).

Original entry on oeis.org

1, 2, 26, 490, 10850, 263010, 6756570, 180732778, 4980586114, 140426468098, 4031581757786, 117456808452906, 3463846465750114, 103200018840208098, 3101624265076611482, 93922235608046966058, 2862850624269320061954, 87768126789137804695298, 2704569471624358219362714
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2023

Keywords

Crossrefs

Cf. A217364, A363006, A363305, A364195.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+3*k+1, n)/(5*n+3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+3*k+1,n)/(5*n+3*k+1).

A365622 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^5 ).

Original entry on oeis.org

1, 10, 150, 2670, 52250, 1086002, 23533790, 525825830, 12026993010, 280220428890, 6627397194022, 158692955007390, 3839595257256330, 93725694152075010, 2305406918530451950, 57085385625207424342, 1421808255906105290210
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A032349, A365843, A365847.
Cf. A363006.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(5*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(5*(n+1),n-k).
G.f.: B^5, where B is the g.f. of A363006.

A371700 G.f. satisfies A(x) = 1 + x * A(x)^6 * (1 + A(x)).

Original entry on oeis.org

1, 2, 26, 482, 10450, 247554, 6208970, 162064322, 4356511138, 119788611458, 3353361311738, 95251219926690, 2738421518770546, 79531905952256642, 2329955712706784682, 68770993359030211458, 2043143866891345880898, 61050342965542475675906
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2024

Keywords

Crossrefs

Cf. A006318, A027307, A144097, A260332, A363006.
Cf. A371678.

Programs

  • PARI
    a(n, r=1, t=6, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+1,n)/(6*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(7*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(6*n,k-1) for n > 0.
Showing 1-7 of 7 results.

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