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.

A273953 E.g.f. satisfies A(x) = Sum_{n>=0} x^n/n! * exp(n/2*x) * A(x)^(n/2).

Original entry on oeis.org

1, 1, 3, 13, 77, 581, 5347, 58213, 732937, 10487737, 168217811, 2990748509, 58397418037, 1242643927357, 28627000014355, 709933328752981, 18859531958840273, 534365880859577777, 16087267158157316323, 512844446937529664173, 17259468942471032848861, 611530055485070740134901, 22755171133646348369448323, 887228501593124485460914373, 36173480392953890421156056665, 1539307965110263598673884269801, 68247672532254821767545000249907
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 77*x^4/4! + 581*x^5/5! + 5347*x^6/6! + 58213*x^7/7! + 732937*x^8/8! + 10487737*x^9/9! + 168217811*x^10/10! + 2990748509*x^11/11! + 58397418037*x^12/12! +...
such that
A(x) = 1 + x*exp(x/2)*A(x)^(1/2) + x^2/2!*exp(x)*A(x) + x^3/3!*exp(3*x/2)*A(x)^(3/2) + x^4/4!*exp(2*x)*A(x)^2 + x^5/5!*exp(5*x/2)*A(x)^(5/2) + x^6/6!*exp(3*x)*A(x)^3 +...
The logarithm of A(x) begins:
log(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1476*x^6/6! + 14728*x^7/7! + 173216*x^8/8! + 2346480*x^9/9! + 35981200*x^10/10! + 616111056*x^11/11! + 11652662880*x^12/12! +...+ A100526(n)*x^n/n! +...
which equals -2*LambertW(-x*exp(x/2)/2).

Links

Crossrefs

Cf. A273954, A100526.

Programs

  • Mathematica
    CoefficientList[Series[4*LambertW[-x/2*E^(x/2)]^2 / (x^2*E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 23 2016 *)
  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A = sum(m=0,n,x^m/m!*exp(m/2*x +x*O(x^n))*A^(m/2)) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    a(n) = sum(k=0, n, k^(n-k)*(k+2)^(k-1)*binomial(n, k))/2^(n-1); \\ Seiichi Manyama, Feb 11 2023

Formula

E.g.f.: 4*LambertW(-x/2*exp(x/2))^2 / (x^2*exp(x)).
E.g.f.: exp( L(x) ) where L(x) = -2*LambertW(-x*exp(x/2)/2) is the e.g.f. of A100526.
a(n) ~ sqrt(1+LambertW(exp(-1)))*n^(n-1)/(2^(n-1)*exp(n-2)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016
From Seiichi Manyama, Feb 11 2023: (Start)
E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(1/2) ).
a(n) = (1/2^(n-1)) * Sum_{k=0..n} k^(n-k) * (k+2)^(k-1) * binomial(n,k). (End)

A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

Original entry on oeis.org

0, 1, 8, 96, 1792, 46080, 1511424, 60325888, 2837970944, 153778913280, 9432255692800, 646039266656256, 48874810528235520, 4047655951598092288, 364221261622538141696, 35384754572803304325120, 3691411033400626898796544, 411569264258973944034361344
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A100526, A216857, A360545.
Cf. A038049, A201595, A214225, A360547.

Programs

  • Maple
    A360548 := proc(n)
    add((2*k)^(n-1)*binomial(n,k),k=1..n) ;
end proc:
seq(A360548(n),n=0..60) ; # R. J. Mathar, Mar 12 2023
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))
  • PARI
    a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = (-1/2) * LambertW(-2*x * exp(2*x)).
a(n) = Sum_{k=1..n} (2*k)^(n-1) * binomial(n,k) = 4^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (LambertW(exp(-1))^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

A360545 E.g.f. satisfies A(x) = x * exp( 3*(x + A(x))/2 ).

Original entry on oeis.org

0, 1, 6, 54, 756, 14580, 358668, 10736712, 378823392, 15395255280, 708217959600, 36380741745744, 2064234271203360, 128214974795177088, 8652900673357097472, 630483717450225530880, 49330027417316557012992, 4124992361928178722764544
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A100526, A216857, A360548.
Cf. A360482, A360544.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-2*lambertw(-3*x/2*exp(3*x/2))/3)))
  • PARI
    a(n) = sum(k=1, n, (3*k/2)^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = (-2/3) * LambertW(-3*x/2 * exp(3*x/2)).
a(n) = Sum_{k=1..n} (3*k/2)^(n-1) * binomial(n,k) = 3^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A273952 E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-x)) ) ) = x, where A(x) = Sum_{n>=1} a(n)*x^n/(2^(n-1)*(n-1)!).

Original entry on oeis.org

1, 1, 1, 4, 77, 736, 2077, 22912, 1197625, 23597056, 350173241, 7161708544, 236337969925, 6751323455488, 122041278706453, 3799979465506816, 298712815532930033, 10872130692620222464, -18153139467375673487, -513247768690867306496, 1172597577739561586096701, 53608628175847428085252096, -748272864671493582192607219, -39715579516869644288006291456, 7586072261976188853665242247977
Offset: 1

Views

Author

Paul D. Hanna, Jun 16 2016

Keywords

Examples

			E.g.f. A(x) = x + x^2/2 + x^3/(2^2*2!) + 4*x^4/(2^3*3!) + 77*x^5/(2^4*4!) + 736*x^6/(2^5*5!) + 2077*x^7/(2^6*6!) + 22912*x^8/(2^7*7!) + 1197625*x^9/(2^8*8!) + 23597056*x^10/(2^9*9!) + 350173241*x^11/(2^10*10!) + 7161708544*x^12/(2^11*11!) + 236337969925*x^13/(2^12*12!) + 6751323455488*x^14/(2^13*13!) + 122041278706453*x^15/(2^14*14!) +...
such that: A( sqrt( A(x^2*exp(-x)) ) ) = x.
Written with reduced fraction coefficients,
A(x) = x + 1/2*x^2 + 1/8*x^3 + 1/12*x^4 + 77/384*x^5 + 23/120*x^6 + 2077/46080*x^7 + 179/5040*x^8 + 239525/2064384*x^9 + 823/6480*x^10 + 350173241/3715891200*x^11 + 109279/1247400*x^12 + 9453518797/78479622144*x^13 + 206034041/1556755200*x^14 + 122041278706453/1428329123020800*x^15 +...
Also, A( sqrt( A(x^2*exp(x)) ) ) = -2*LambertW(-x/2*exp(x/2)) where
A( sqrt( A(x^2*exp(x)) ) ) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1476*x^6/6! +...+ A100526(n)*x^n/n! +...

Crossrefs

Cf. A100526.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = serreverse( sqrt( subst(A,x,x^2*exp(-x +x*O(x^n))) ) ) ); (n-1)!*2^(n-1) * polcoeff(A,n)}
for(n=1,30, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / (2^(n-1)*(n-1)!) satisfies:
(1) A( sqrt( A(x^2*exp(x)) ) ) = -2*LambertW(-x/2*exp(x/2)).
(2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-x)) ) ).
Showing 1-4 of 4 results.

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