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 1-5 of 5 results.

A052750 a(n) = (2*n + 1)^(n - 1).

Original entry on oeis.org

1, 1, 5, 49, 729, 14641, 371293, 11390625, 410338673, 16983563041, 794280046581, 41426511213649, 2384185791015625, 150094635296999121, 10260628712958602189, 756943935220796320321, 59938945498865420543457
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

a(n+1) is the number of labeled incomplete ternary trees on n vertices in which each left child has a larger label than its parent. - Brian Drake, Jul 28 2008
Put a(0) = 1. For n > 0, let x(n,k) = 2*cos((2*k-1)*Pi/(2*n+1)), k=1..n. Define the recurrences S(n;0,x(n,k)) = 1, S(n;1,x(n,k)) = x(n,k), S(n;r,x(n,k)) = x(n,k)*S(n;r-1,x(n,k)) - S(n;r-2,x(n,k)), r > 1 an integer, k=1..n. CONJECTURE: For n > 0, a(n) = Product_{k=1..n} (Sum_{m=0..n-1} S(n;2*m,x(n,k))^2). - L. Edson Jeffery, Sep 11 2013
From Wolfdieter Lang, Dec 16 2013: (Start)
Discriminants of the first difference of Chebyshev S-polynomials.
The coefficient table for the first difference polynomials P(n, x) = S(n, x) - S(n-1, x), n >= 0, S(-1, x) = 0, with the Chebyshev S polynomials (see A049310), is given in A130777.
For the discriminant of a polynomial in terms of the square of a determinant of a Vandermonde matrix build from the zeros of the polynomial see, e.g., A127670.
For the proof that D(n) := discriminant(P(n,x)) = (2*n + 1)^(n - 1), n >= 1, use the formula given e.g., in the Rivlin reference, p. 218, Theorem 5.13, eq. (5.3), namely D(n) = (-1)^(n*(n-1)/2)*Product_{j=1..n} P'(n, x(n,j)), with the zeros x(n,j) = -2*cos(2*Pi*j/(2*n+1)) of P(n, x) (see A130777). P'(n, x(n,j)) = (2*n+1)*P(n-1, x(n,j))/(2*sin(Pi*j/(2*n+1))*2*cos(Pi*j/(2*n+1)))^2. P(n-1, x(n,j)) = (-1)^(n+j)*2*cos(Pi*j/(2*n+1)). Product_{j=1..n} 2*sin(Pi*j/(2*n+1)) = 2*n+1 (see the Oct 10 2013 formula in A005408. Product_{j=1..n} 2*cos(Pi*j/(2*n+1)) = 1, because S(2*n, 0) = (-1)^n.
(End)
a(n) is the number of labeled 2-trees with n+2 vertices, rooted at a given edge. - Nikos Apostolakis, Nov 30 2018
a(n) is also the number of 2-trees with n labeled triangles and with a distinguished oriented edge. - Nikos Apostolakis, Dec 14 2018

Examples

			Discriminant: n=4: P(4, x) = 1 + 2*x - 3*x^2 - x^3 + x^4 with the zeros x[1] = -2*cos((2/9)*Pi), x[2] = -2*cos((4/9)*Pi), x[3] = 1, x[4] = 2*cos((1/9)*Pi). D(4) = (Det(Vandermonde(4,[x[1],x[2],x[3],x[4]])))^2 = 729 = a(4). - _Wolfdieter Lang_, Dec 16 2013

References

  • L. W. Beineke, and J. W. Moon, Several proofs of the number of labelled 2-dimensional trees, In "Proof Techniques in Graph Theory" (F. Harary editor). Academic Press, New York, 1969, pp. 11-20.
  • Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Links

Crossrefs

Cf. A127670, A130777, A000169, A052752, A052774, A052782, A000272, A001705.

Programs

  • GAP
    List([0..20],n->(2*n+1)^(n-1)); # Muniru A Asiru, Dec 05 2018
  • Magma
    [(2*n+1)^(n-1) : n in [0..20]]; // Wesley Ivan Hurt, Jan 20 2017
  • Maple
    spec := [S,{B=Prod(Z,S,S),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    max = 16; (Series[Exp[-1/2*ProductLog[-2*x]], {x, 0, max}] // CoefficientList[#, x] & ) * Range[0, max]! (* Jean-François Alcover, Jun 20 2013 *)
Table[(2*n+1)^(n-1),{n,0,20}] (* Harvey P. Dale, Jul 14 2025 *)
  • PARI
    a(n)=(2*n+1)^(n-1) \\ Charles R Greathouse IV, Nov 20 2011
  • PARI
    {a(n)=local(A=1+x);for(i=1,21,A=sqrt(1+2*sum(n=1,21,x^(2*n-1)/(2*n-1)!*A^(4*n-1))+x*O(x^n)));n!*polcoeff(A,n)} \\ Paul D. Hanna, Sep 07 2012
  • Python
    for n in range(0, 20): print((2*n + 1)**(n - 1), end=', ') # Stefano Spezia, Dec 01 2018

Formula

E.g.f.: exp(-1/2*W(-2*x)), where W is Lambert's W function.
E.g.f. satisfies: A(x) = sqrt(1 + 2*Sum_{n>=1} x^(2*n-1)/(2*n-1)! * A(x)^(4*n-1)). - Paul D. Hanna, Sep 07 2012
E.g.f. satisfies: A(x) = 1/A(-x*A(x)^4). - Paul D. Hanna, Sep 07 2012
a(n) = discriminant of P(n,x) = S(n,x) - S(n-1,x), n >= 1, with the Chebyshev S polynomials from A049310. For the proof see the comment above. a(n) is also the discriminant of S(n,x) + S(n-1,x) = (-1)^n*(S(n,-x) - S(n-1,-x)). - Wolfdieter Lang, Dec 16 2013
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + ... satisfies:
1) A(x*exp(-2*x)) = exp(x) = 1/A(-x*exp(2*x));
2) A^2(x) = 1/x*series reversion(x*exp(-2*x));
3) A(x^2) = 1/x*series reversion(x*exp(-x^2));
4) A(x) = exp(x*A(x)^2). (End)
E.g.f.: sqrt(-LambertW(-2*x)/(2*x)). - Vaclav Kotesovec, Dec 07 2014
Related to A001705 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^2*log(1 + x) ) = series reversion(x - 5*x^2/2! + 26*x^3/3! - 154*x^4/4! + ...). Cf. A000272, A052752, A052774, A052782. - Peter Bala, Jun 15 2016
From Peter Bala, Dec 13 2022: (Start)
The e.g.f. A(x) = 1/x * series reversion of x^2/T(x), where the tree function T(x) = Sum_{n >= 1} n^(n-1)*x^n/n!. See A000169.
For c in C, A(x)^c = 1 + Sum_{n >= 1} c*(2*n + c)^(n-1)*x^n/n!.
First derivative A'(x) = A(x)^3/(1 - 2*x*A(x)^2).
Series reversion of (1 - A(-z)) = -log(1 - z)/(1 - z)^2 is the e.g.f. of A001705.
1/z * series reversion of z/A(z) = 1 + z + 7*z^2/2! + (10^2)*z^3/3! + (13^3)*z^4/4! + ... is the e.g.f. of A052752.
1/z * series reversion of z/A(z^2) = 1 + z^2 + 9*z^4/2! + (13^2)*z^6/3! + (17^3)*z^8/4! + ... = Sum_{n >= 0} A052774(n)*z^(2*n)/n!.
1/z * series reversion of z/A(z^3) = 1 + z^3 + 11*z^6/2! + (16^2)*z^9/3! + (21^3)*z^12/4! + ... = Sum_{n >= 0} A052782(n)*z^(3*n)/n!.
1/z * series reversion of z/A(z)^2 = A(2*z) = 2*Sum_{n >= 0} (4*n + 2)^(n-1)*z^n/n!.
1/z * series reversion of z/A(z)^k = k*Sum_{n >= 0} ((k+2)*n + k)^(n-1)*z^n/n!. (End)
a(n) = Sum_{k=1..n} (-1)^(n-k)*(n+k)^(n-1)*binomial(n,k-1), a(0)=1. - Vladimir Kruchinin, Aug 14 2025

Extensions

Better description from Vladeta Jovovic, Sep 02 2003

A052752 a(n) = (3*n+1)^(n-1).

Original entry on oeis.org

1, 1, 7, 100, 2197, 65536, 2476099, 113379904, 6103515625, 377801998336, 26439622160671, 2064377754059776, 177917621779460413, 16777216000000000000, 1718264124282290785243, 189937030341242876870656, 22539340290692258087863249, 2857942574656970690381479936
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A000169, A052750, A052774, A052782, A000272, A001711.

Programs

  • Magma
    [(3*n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017
  • Maple
    spec := [S,{B=Prod(S,S,S,Z),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Table[(3n+1)^(n-1),{n,0,20}] (* Harvey P. Dale, Aug 14 2015 *)
With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-3*x]/3], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 16 2017 *)
  • PARI
    for(n=0,50, print1((3*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017
  • PARI
    x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-3*x)/3))) \\ G. C. Greubel, Nov 16 2017

Formula

E.g.f.: exp(-(1/3)*LambertW(-3*x)).
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + ... satisfies:
1) A(x*exp(-3*x)) = exp(x) = 1/A(-x*exp(3*x));
2) A^3(x) = 1/x*series reversion(x*exp(-3*x));
3) A(x^3) = 1/x*series reversion(x*exp(-x^3));
4) A(x) = exp(x*A(x)^3);
5) A(x) = 1/A(-x*A(x)^6). (End)
E.g.f.: (-LambertW(-3*x)/(3*x))^(1/3). - Vaclav Kotesovec, Dec 07 2014
Related to A001711 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^3*log(1 + x) ) = series reversion(x - 7*x^2/2! + 47*x^3/3! - 342*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016
a(n) = Sum_{k=1..n} (-1)^(n-k)*(2n+k)^(n-1)*binomial(n,k-1), a(0)=1. - Vladimir Kruchinin, Aug 14 2025

Extensions

Better description from Vladeta Jovovic, Sep 02 2003

A052782 a(n) = (5*n+1)^(n-1).

Original entry on oeis.org

1, 1, 11, 256, 9261, 456976, 28629151, 2176782336, 194754273881, 20047612231936, 2334165173090451, 303305489096114176, 43513917611435838661, 6831675453247426400256, 1165087474585497590531111, 214481724045177216015794176, 42391158275216203514294433201
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A000169, A052750, A052752, A052774, A000272, A001721.

Programs

  • Magma
    [(5*n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017
  • Maple
    spec := [S,{S=Set(B),B=Prod(Z,S,S,S,S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-5*x]/5], {x, 0, nmax}], x]*Range[0, nmax]!] (* or *) Table[(5*n+1)^(n-1), {n,0,50}] (* G. C. Greubel, Nov 16 2017 *)
  • PARI
    for(n=0,50, print1((5*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017
  • PARI
    x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-5*x)/5))) \\ G. C. Greubel, Nov 16 2017

Formula

E.g.f.: exp(-1/5*LambertW(-5*x)).
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + ... satisfies:
1) A(x*exp(-5*x)) = exp(x) = 1/A(-x*exp(5*x));
2) A^5(x) = 1/x*series reversion(x*exp(-5*x));
3) A(x^5) = 1/x*series reversion(x*exp(-x^5));
4) A(x) = exp(x*A(x)^5);
5) A(x) = 1/A(-x*A(x)^10). (End)
E.g.f.: (-LambertW(-5*x)/(5*x))^(1/5). - Vaclav Kotesovec, Dec 07 2014
Related to A001721 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^5*log(1 + x) ) = series reversion(x - 11*x^2/2! + 107*x^3/3! - 1066*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016

Extensions

Better description from Vladeta Jovovic, Sep 02 2003

A274267 a(n) = (4*n - 1)^(n-1).

Original entry on oeis.org

1, 7, 121, 3375, 130321, 6436343, 387420489, 27512614111, 2251875390625, 208728361158759, 21611482313284249, 2472159215084012303, 309629344375621415601, 42141982597572021484375, 6193386212891813387462761, 977480813971145474830595007, 164890958756244164895763202881
Offset: 1

Views

Author

Peter Bala, Jun 19 2016

Keywords

Comments

Compare with A052774.

Links

Crossrefs

Cf. A000169, A052774, A274265, A274266, A274268, A274269.

Programs

  • Magma
    [(4*n-1)^(n-1): n in [1..25]]; // Vincenzo Librandi, Jun 20 2016
  • Maple
    A274267 := n -> (4*n - 1)^(n-1):
seq(A274267(n), n = 1..20);
  • Mathematica
    Table[(4*n-1)^(n-1), {n,1,25}] (* G. C. Greubel, Jun 19 2016 *)
  • PARI
    for(n=1,30, print1((4*n-1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017

Formula

E.g.f. A(x) = 1 - exp(-1/4*T(4*x)) = x + 7*x^2/2! + 11^2*x^3/3! + 15^3*x^4/4! + 19^4*x^5/5! + ..., where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
A(x) = series reversion( (1 - x)^4*log(1/(1 - x)) ). See A274268.
1 - A(x) = exp(-x/(1 - A(x))^4) = exp(-x/(exp(-4*x/(exp(-4*x/ ...))))).
1 - A(-x*exp(4*x)) = exp(x) = 1/(1 - A(x*exp(-4*x))).
1/(1 - A(x)) = Sum_{n >= 0} (4*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052774.

A385060 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/A(-x*A(x))^3).

Original entry on oeis.org

1, 1, 9, 43, 125, -6279, -412025, -9060911, -98234103, 10633082401, 1397628908171, 76301056937865, 3570677332342477, -21017427583370399, -28544077526642620977, -3624219130398256821119, -403594553235196466813935, -26214785940427662614214975, -249746440937109731554641389
Offset: 0

Views

Author

Seiichi Manyama, Jun 16 2025

Keywords

Crossrefs

Column k=1 of A385063.
Cf. A052774, A384984.

Programs

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

Formula

See A385063.
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