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.

A370276 Self-convolution of A138020.

Original entry on oeis.org

1, 4, 16, 72, 352, 1816, 9728, 53584, 301568, 1726488, 10022912, 58864240, 349102080, 2087772784, 12576358400, 76237953440, 464736354304, 2847019090712, 17518413479936, 108224749140784, 670996707147776, 4173817417204944, 26040046909915136, 162905940337309792, 1021700454913933312
Offset: 0

Views

Author

Alexander Burstein, Feb 13 2024

Keywords

Crossrefs

Cf. A138020, A000984.

Programs

  • Maple
    A370276 := proc(n)
    add( A138020(i)*A138020(n-i),i=0..n) ;
end proc:
seq(A370276(n),n=0..80) ; # R. J. Mathar, Sep 27 2024
  • Mathematica
    CoefficientList[(InverseSeries[Series[x Sqrt[(1-2x)/(1+2x)],{x,0,25}]])^2/x^2,x]

Formula

G.f.: A(x) = F(x)^2, where F(x) is the g.f. of A138020.
G.f.: (A(x)-1)/(A(x)+1) = 2*x*sqrt(A(x)) = 2*x*F(x).
G.f.: A(-x*A(x)) = 1/A(x).
G.f.: A(x) = 1 + 4*x*A(x)*B(x^2*A(x)), where B(x) is the g.f. of the central binomial coefficients A000984.
D-finite with recurrence (n-1)*(n+2)*(5*n-12)*a(n) +4*(-55*n^3+242*n^2-316*n+120)*a(n-2) -16*(n-3)*(n-4)*(5*n-2)*a(n-4)=0. - R. J. Mathar, Sep 27 2024

A385501 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arctanh(x)) ).

Original entry on oeis.org

1, 1, 3, 18, 165, 2040, 31815, 599760, 13268745, 337115520, 9674678475, 309554784000, 10927053262125, 421849524096000, 17682153623909775, 799730490214656000, 38820939579369572625, 2013202580708487168000, 111081054630965602057875, 6497703571257963896832000
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2025

Keywords

Crossrefs

Cf. A385500, A385502.
Cf. A001147, A111594, A138020, A227466, A385426.

Programs

  • Mathematica
    nmax=20; CoefficientList[(1/x) *InverseSeries[Series[x * Exp[-ArcTanh[x]],{x,0,nmax}],x] ,x]Range[0,nmax-1]! (* Stefano Spezia, Jul 01 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, binomial(n, k)*binomial(n/2+k+1/2, n)/(n+2*k+1));

Formula

E.g.f. A(x) satisfies A(x) = exp( arctanh(x*A(x)) ).
E.g.f. A(x) satisfies A(x) = sqrt( (1+x*A(x))/(1-x*A(x)) ).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * A111594(n,k).
a(n) = n!/2^n * A138020(n) = n! * Sum_{k=0..n} binomial(n,k) * binomial(n/2+k+1/2,n)/(n+2*k+1).

A379382 G.f. A(x) satisfies A(x) = sqrt( (1 + 2*x*A(x))/(1 - 2*x*A(x)^2) ).

Original entry on oeis.org

1, 2, 8, 48, 336, 2560, 20608, 172416, 1484288, 13062144, 116977664, 1062600704, 9767067648, 90673700864, 848971661312, 8007542571008, 76014137180160, 725681289822208, 6962697126019072, 67105309925048320, 649362348326256640, 6306663216709632000
Offset: 0

Views

Author

Seiichi Manyama, Dec 22 2024

Keywords

Crossrefs

Cf. A138020, A379383.
Cf. A151374, A379327, A379330.
Cf. A106228.

Programs

  • PARI
    a(n) = 2^n*sum(k=0, n, binomial(n/2+3*k/2-1/2, k)*binomial(n/2+k/2+1/2, n-k)/(n+k+1));

Formula

a(n) = 2^n * Sum_{k=0..n} binomial(n/2+3*k/2-1/2,k) * binomial(n/2+k/2+1/2,n-k)/(n+k+1).
a(n) = 2^n * A106228(n).

A379383 G.f. A(x) satisfies A(x) = sqrt( (1 + 2*x*A(x))/(1 - 2*x*A(x)^3) ).

Original entry on oeis.org

1, 2, 10, 80, 750, 7680, 83252, 939008, 10905942, 129548288, 1566565452, 19220267008, 238662840780, 2993651974144, 37876206019560, 482802294325248, 6194365014836582, 79930063134392320, 1036640587694252380, 13505632613590630400, 176673045664669396132, 2319654465118014537728
Offset: 0

Views

Author

Seiichi Manyama, Dec 22 2024

Keywords

Crossrefs

Cf. A138020, A379382.
Cf. A078531, A379328, A379331.

Programs

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

Formula

a(n) = 2^n * Sum_{k=0..n} binomial(n/2+2*k-1/2,k) * binomial(n/2+k+1/2,n-k)/(n+2*k+1).
a(n) = 2^n * Sum_{k=0..n} binomial(n,k) * binomial(n/2+2*k+1/2,n)/(n+4*k+1).
Showing 1-4 of 4 results.

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