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 1-10 of 14 results. Next

A273954 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * exp(n*x) * A(x)^n.

Original entry on oeis.org

1, 1, 5, 37, 393, 5481, 95053, 1975821, 47939601, 1330923601, 41629292181, 1448989481589, 55561575788953, 2327512861252281, 105767732851318749, 5182512561142513501, 272391086209524010017, 15287595381259195453089, 912525533175190887597349, 57726267762799335649572549
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 393*x^4/4! + 5481*x^5/5! + 95053*x^6/6! + 1975821*x^7/7! + 47939601*x^8/8! + 1330923601*x^9/9! + 41629292181*x^10/10! + 1448989481589*x^11/11! + 55561575788953*x^12/12! +...
such that
A(x) = 1 + x*exp(x)*A(x) + x^2/2!*exp(2*x)*A(x)^2 + x^3/3!*exp(3*x)*A(x)^3 + x^4/4!*exp(4*x)*A(x)^4 + x^5/5!*exp(5*x)*A(x)^5 + x^6/6!*exp(6*x)*A(x)^6 +...
The logarithm of A(x) begins:
log(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! + 600698880*x^9/9! + 18422374400*x^10/10! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).

Links

Crossrefs

Cf. A273953, A216857, A357247, A360176 (column 1 unsigned).
Cf. A161628, A244119.

Programs

  • Maple
    A273954 := n -> add(binomial(n, j) * j^(n - j) * (j + 1)^(j - 1), j = 0..n):
seq(A273954(n), n = 0..24); # Peter Luschny, Jan 29 2023
  • Mathematica
    CoefficientList[Series[-LambertW[-x*E^x] / (x*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*x +x*O(x^n))*A^m) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    x='x+O('x^50); Vec(serlaplace(-lambertw(-x*exp(x))/(x*exp(x)))) \\ G. C. Greubel, Nov 16 2017
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(x))^k/k!))) \\ Seiichi Manyama, Feb 08 2023

Formula

E.g.f.: -LambertW(-x*exp(x)) / (x*exp(x)). [corrected by Vaclav Kotesovec, Jun 23 2016]
E.g.f.: exp( L(x) ) where L(x) = -LambertW(-x*exp(x)) is the e.g.f. of A216857.
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016
E.g.f.: A(x) = exp(x*exp(x)*A(x)). - Alexander Burstein, Aug 11 2018
From Peter Luschny, Jan 29 2023: (Start)
a(n) = Sum_{j=0..n} binomial(n, j) * j^(n - j) * (j + 1)^(j - 1).
a(n) = Sum_{k=0..n} (-1)^k*A161628(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*A244119(n, k). (End)

A349562 Number of labeled rooted forests with 2-colored leaves.

Original entry on oeis.org

1, 2, 8, 56, 576, 7872, 134656, 2771456, 66744320, 1842237440, 57354338304, 1988721131520, 76015173369856, 3175757373243392, 143980934947930112, 7040807787705663488, 369414622819764928512, 20700889684976244621312, 1233951687316746828513280, 77963762014950356953333760
Offset: 0

Views

Author

Alexander Burstein, Nov 22 2021

Keywords

Comments

a(n) is the number of labeled trees on vertices 0,1,...,n rooted at 0, where all leaves have 2 colors (except the singleton tree 0 has only 1 color).

Examples

			a(2)=8 counts trees 0-1-2B, 0-1-2R, 0-2-1B, 0-2-1R, 1B-0-2B, 1B-0-2R, 1R-0-2B, 1R-0-2R (where B and R stand for colors Blue and Red).

Links

Crossrefs

Cf. A000007, A007889, A216857, A038049, A201595,

Programs

  • Mathematica
    CoefficientList[u/.AsymptoticSolve[u-E^(x(1+u))==0,u->1,{x,0,24}][[1]],x]Factorial/@Range[0,24]
nmax = 20; CoefficientList[Series[-LambertW[-x*Exp[x]]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 25 2021 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k+1)^(k-1)*x^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Nov 26 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(n-1).
a(n) = A216857(n+1)/(n+1).
a(n) = A038049(n+1)/(n+1) for n>=1, and a(0) = A038049(1)/2.
a(n) = 2*A201595(n) - A000007(n).
E.g.f. satisfies: A(x) = e^(x*(1 + A(x))).
E.g.f. satisfies: A(-x*A(x)) = 1/A(x).
From Vaclav Kotesovec, Nov 25 2021: (Start)
E.g.f.: -LambertW(-x*exp(x))/x.
a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n+1)).
(End)
From Seiichi Manyama, Nov 26 2021: (Start)
G.f.: Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1).
a(n) = 2^n * A007889(n). (End)

A357247 E.g.f. satisfies A(x) * log(A(x)) = x * exp(-x).

Original entry on oeis.org

1, 1, -3, 13, -103, 1241, -19691, 384805, -8918351, 238966705, -7265920339, 247123552061, -9295263915191, 383095792217737, -17167554097899323, 831082449069928021, -43221681697593767071, 2403219105771778162529, -142263939562414917333155
Offset: 0

Views

Author

Seiichi Manyama, Sep 19 2022

Keywords

Links

Crossrefs

Cf. A177885, A216857, A357243, A357246, A359759 (column 1).

Programs

  • Maple
    A357247 := n -> (-1)^(n - 1) * add(binomial(n, j) * (j - 1)^(j - 1) * j^(n - j), j = 0..n): seq(A357247(n), n = 0..18); # Peter Luschny, Jan 28 2023
  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/(Exp[x]*A[x])] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*exp(-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*exp(-x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(x*exp(-x)/lambertw(x*exp(-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=1, j, (-k)^(j-1)*binomial(j, k))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

E.g.f. satisfies A(x) * log(A(x)) - x * exp(-x) = 0.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (x * exp(-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(x * exp(-x)) ).
E.g.f.: A(x) = x * exp(-x)/LambertW(x * exp(-x)).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * A216857(k) * binomial(n-1,k-1) * a(n-k).
a(n) = (-1)^(n - 1) * Sum_{j=0..n} binomial(n, j) * (j - 1)^(j - 1) * j^(n - j). - Peter Luschny, Jan 28 2023
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n+1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jan 28 2023

A274275 E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-2*x)) ) ) = x.

Original entry on oeis.org

1, 2, 6, 40, 400, 4656, 62944, 1046144, 20274048, 438238720, 10529132416, 280439144448, 8185848206848, 259202608222208, 8855252721592320, 324989707586830336, 12748521382531956736, 532098814401540587520, 23547710868033300004864, 1101540715832518509854720, 54307901369414002580422656, 2814303585179538846791237632, 152935335939643406489642008576, 8696644113583584719506275041280, 516469893784923819203984490496000
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2016

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 400*x^5/5! + 4656*x^6/6! + 62944*x^7/7! + 1046144*x^8/8! + 20274048*x^9/9! + 438238720*x^10/10! + 10529132416*x^11/11! + 280439144448*x^12/12! + 8185848206848*x^13/13! + 259202608222208*x^14/14! + 8855252721592320*x^15/15! + 324989707586830336*x^16/16! +...
such that A( sqrt( A(x^2*exp(-2*x)) ) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
sqrt( A(x^2*exp(-2*x)) ) = x - 2*x^2/2! + 6*x^3/3! - 40*x^4/4! + 320*x^5/5! - 2976*x^6/6! + 35392*x^7/7! - 538112*x^8/8! + 9931392*x^9/9! - 211790080*x^10/10! + 5059784576*x^11/11! - 132643057152*x^12/12! + 3761875287040*x^13/13! - 114501941915648*x^14/14! + 3725395402721280*x^15/15! - 129324055589257216*x^16/16! +...+ A274277(n)*x^n/n! +...
Compare the above series reversion to the following series:
A(x)^2 * exp(-2*A(x)) = x^2 - 2*x^4/2! + 6*x^6/3! - 40*x^8/4! + 320*x^10/5! - 2976*x^12/6! + 35392*x^14/7! - 538112*x^16/8! + 9931392*x^18/9! +...
where A( A(x)^2 * exp(-2*A(x)) ) = x^2.
The e.g.f. A(x) is related to the LambertW function by the composition:
A( sqrt( A(x^2*exp(2*x)) ) ) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).

Links

Crossrefs

Cf. A274276, A216857, A274277.
Cf. variants: A274393, A274394, A274395.

Programs

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

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) A( sqrt( A(x^2*exp(2*x)) ) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-2*x)) ) ).
(3) A( A(x)^2 * exp(-2*A(x)) ) = x^2.
(4) A(-A(x)^2 * exp(-2*A(x)) ) = -LambertW(x^2*exp(-x^2)).
a(n)/n! ~ c * d^n / n^(3/2), where d = 2.52462188117..., c = 0.36965356... . - Vaclav Kotesovec, Jun 24 2016

A274394 E.g.f. A(x) satisfies: A( A( x^4*exp(-4*x) )^(1/4) ) = x.

Original entry on oeis.org

1, 2, 9, 64, 595, 7416, 111979, 1989632, 40695561, 941667040, 24323649361, 693818707968, 21661372820971, 734712173277824, 26902827107293635, 1057724890214957056, 44442356900221356241, 1987370544970750468608, 94240073170115929379161, 4723448516579307027169280, 249510355552473169494452931, 13854414947224528743034304512, 806733172355775780726416256859
Offset: 1

Views

Author

Paul D. Hanna, Jun 21 2016

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 595*x^5/5! + 7416*x^6/6! + 111979*x^7/7! + 1989632*x^8/8! + 40695561*x^9/9! + 941667040*x^10/10! + 24323649361*x^11/11! + 693818707968*x^12/12! + 21661372820971*x^13/13! + 734712173277824*x^14/14! + 26902827107293635*x^15/15! + 1057724890214957056*x^16/16! +...
such that A( A( x^4*exp(-4*x) )^(1/4) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
A( x^4*exp(-4*x) )^(1/4) = x - 2*x^2/2! + 3*x^3/3! - 4*x^4/4! + 35*x^5/5! - 906*x^6/6! + 15757*x^7/7! - 210008*x^8/8! + 2464569*x^9/9! - 32810410*x^10/10! + 671239811*x^11/11! - 18224632812*x^12/12! + 496597765963*x^13/13! - 12681217528994*x^14/14! + 320976165059565*x^15/15! +...
Compare the above series reversion to the following series:
A(x)^4 * exp(-4*A(x)) = x^4 - 2*x^8/2! + 3*x^12/3! - 4*x^16/4! + 35*x^20/5! - 906*x^24/6! + 15757*x^28/7! - 210008*x^32/8! + 2464569*x^36/9! - 32810410*x^40/10! +...
where A( A(x)^4 * exp(-4*A(x)) ) = x^4.
The e.g.f. A(x) is related to the LambertW function by the composition:
A( A(x^4*exp(4*x))^(1/4) ) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).

Links

Crossrefs

Cf. A274275, A274393, A274395.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^4*exp(-4*x +x*O(x^n)))^(1/4) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

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

A360432 E.g.f. satisfies A(x) = x * exp(A(x) + x^2).

Original entry on oeis.org

0, 1, 2, 15, 112, 1225, 16896, 283759, 5623808, 128431377, 3321026560, 95915951791, 3060250165248, 106896447626137, 4057412577591296, 166284754020913935, 7318183421113532416, 344228133020323687201, 17233838271273426223104, 915000759922243030582735
Offset: 0

Views

Author

Seiichi Manyama, Feb 07 2023

Keywords

Links

Crossrefs

Cf. A216857, A360433.
Cf. A052318.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-x*exp(x^2)))))

Formula

E.g.f.: -LambertW( -x*exp(x^2) ).
a(n) ~ sqrt(1+LambertW(2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (LambertW(2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Feb 07 2023

A360481 E.g.f. satisfies A(x) = x * exp(x + 2 * A(x)).

Original entry on oeis.org

0, 1, 6, 63, 1044, 23805, 692118, 24482115, 1020584232, 49000005945, 2662853279850, 161586078510879, 10830019921469532, 794577001293803637, 63339899145968483262, 5451312770064188283195, 503784284643602483767632, 49757423537114340032969073
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Links

Crossrefs

Cf. A216857, A360482, A360483, A360484.
Cf. A360473.

Programs

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

Formula

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

A360482 E.g.f. satisfies A(x) = x * exp(x + 3 * A(x)).

Original entry on oeis.org

0, 1, 8, 120, 2848, 92960, 3868224, 195810496, 11680512512, 802445898240, 62396469222400, 5417515922441216, 519519435065020416, 54535504354085687296, 6219954774471102242816, 765903524713482618101760, 101269330068289021683564544, 14310318526812295078276628480
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Links

Crossrefs

Cf. A216857, A360481, A360483, A360484.

Programs

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

Formula

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

A360483 E.g.f. satisfies A(x) = x * exp(x - 2 * A(x)).

Original entry on oeis.org

0, 1, -2, 15, -172, 2685, -53226, 1281091, -36296408, 1183527225, -43660076950, 1797823266591, -81746462498724, 4068086310006901, -219929012455113794, 12835335232410655035, -804287930238495495856, 53858337558670992931185
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Links

Crossrefs

Cf. A216857, A360481, A360482, A360484.

Programs

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

Formula

E.g.f.: A(x) = LambertW(2*x * exp(x))/2.
a(n) = Sum_{k=1..n} (-2)^(k-1) * k^(n-1) * binomial(n,k).

A360484 E.g.f. satisfies A(x) = x * exp(x - 3 * A(x)).

Original entry on oeis.org

0, 1, -4, 48, -896, 22880, -743232, 29337280, -1363752448, 72979407360, -4419108684800, 298730433250304, -22300928914403328, 1822195561572585472, -161756111552270491648, 15501595224386724126720, -1595092357302221461127168, 175405731698165304882495488
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Links

Crossrefs

Cf. A216857, A360481, A360482, A360483.

Programs

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

Formula

E.g.f.: A(x) = LambertW(3*x * exp(x))/3.
a(n) = Sum_{k=1..n} (-3)^(k-1) * k^(n-1) * binomial(n,k).
Showing 1-10 of 14 results. Next

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