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 12 results. Next

A218300 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(2*x)*cosh(2*x).

Original entry on oeis.org

1, 2, 12, 104, 1216, 18112, 329600, 7108096, 177549312, 5046554624, 160947232768, 5694342479872, 221410157133824, 9387011838312448, 431051678297358336, 21316106766591721472, 1129526392342026649600, 63855305138514241257472, 3836490516381680506241024
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2012

Keywords

Comments

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

Examples

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 104*x^3/3! + 1216*x^4/4! + 18112*x^5/5! +...
where
A(x) = cosh(2*x) + 2*3^0*cosh(3*x)*x + 2*4^1*cosh(4*x)*x^2/2! + 2*5^2*cosh(5*x)*x^3/3! + 2*6^3*cosh(6*x)*x^4/4! + 2*7^4*cosh(7*x)*x^5/5! +...

Links

Crossrefs

Cf. A201595, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310, A217900.
Cf. A202357.

Programs

  • Mathematica
    nmin = 0; nmax = 18; sol = {a[0] -> 1}; nsol = Length[sol];
Do[A[x_] = Sum[a[k] x^k/k!, {k, 0, n}] /. sol; eq = CoefficientList[ A[x/(Exp[x] Cosh[x])] - Exp[2x] Cosh[2x] + O[x]^(n+1), x][[nsol+1;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol+1, nmax}];
a /@ Range[nmin, nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(2*R)*cosh(2*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,2*(k+2)^(k-1)*cosh((k+2)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: A(x) = Sum_{n>=0} 2*(n+2)^(n-1) * cosh((n+2)*x) * x^n/n!.
E.g.f.: A(x) = 1 + Sum_{n>=0} 2*(n+2)^(n-1) * sinh((n+2)*x) * x^n/n!.
a(n) ~ c * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/LambertW(exp(-1))^2 = 14.5815783688217906961670551786416446... . - Vaclav Kotesovec, Jul 13 2014, updated Jun 10 2019
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 2*x - 2*LambertW(-x * exp(x)) ).
a(n) = Sum_{k=0..n} (k+2)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + Sum_{k>=0} (k+2)^(k-1) * x^k/(1 - (k+2)*x)^(k+1). (End)

A202348 Decimal expansion of x satisfying x = exp(x-2).

Original entry on oeis.org

1, 5, 8, 5, 9, 4, 3, 3, 9, 5, 6, 3, 0, 3, 9, 3, 6, 2, 1, 5, 3, 3, 9, 5, 3, 4, 1, 9, 8, 7, 5, 1, 3, 8, 9, 3, 9, 4, 9, 6, 2, 8, 6, 8, 5, 6, 2, 2, 3, 6, 0, 3, 3, 0, 4, 3, 5, 1, 4, 9, 9, 1, 0, 4, 5, 1, 8, 1, 5, 9, 1, 8, 7, 8, 1, 6, 8, 2, 9, 9, 9, 4, 8, 9, 6, 5, 4, 3, 2, 8, 3, 0, 8, 6, 9, 5, 9, 8, 3
Offset: 0

Views

Author

Clark Kimberling, Dec 20 2011

Keywords

Comments

For many choices of u and v, there is just one value of x satisfying x = exp(u*x+v). Guide to related sequences, with graphs included in Mathematica programs:
u v x
----- -- -------
1 -2 A202348
1 -3 A202494
-1 -1 A202357
-1 -2 A202496
-2 -2 A202497
-2 0 A202498
-3 0 A202499
-Pi 0 A202500
-Pi/2 0 A202501
-2*Pi -1 A202495
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v) = 0. We call the graph of z = g(u,v) an implicit surface of f.
For an example related to this sequence, take f(x,u,v) = x - exp(u*x+v) and g(u,v) = a nonzero solution x of f(x,u,v) = 0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
Actually there are two solutions to x = exp(x-2). This sequence gives the lesser one, x = -LambertW(-exp(-2)), and A226572 gives the greater one, x = -LambertW(-1,-exp(-2)) = 3.14619322062... - Jianing Song, Dec 30 2018

Examples

			x = 0.158594339563039362153395341987513893949...

Links

Crossrefs

Cf. A202320, A226572.

Programs

  • Mathematica
    (* Program 1: A202348 *)
u = 1; v = -2;
f[x_] := x; g[x_] := E^(u*x + v)
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .15, .16}, WorkingPrecision -> 110]
RealDigits[r]  (* A202348 *)
(* Program 2: implicit surface of x=e^(ux+v) *)
f[{x_, u_, v_}] := x - E^(u*x + v);
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, .3}]}, {v, 1, 5}, {u, -5, -.1}];
ListPlot3D[Flatten[t, 1]] (* for A202348 *)
RealDigits[-ProductLog[-1/E^2], 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *)
  • PARI
    solve(x=0,1,exp(x-2)-x) \\ Charles R Greathouse IV, Feb 26 2013

Formula

Equals -LambertW(-exp(-2)) = 2 - A202320. - Jianing Song, Dec 30 2018

Extensions

Digits from a(93) on corrected by Jean-François Alcover, Feb 26 2013

A202322 Decimal expansion of x satisfying x+2=exp(-x).

Original entry on oeis.org

4, 4, 2, 8, 5, 4, 4, 0, 1, 0, 0, 2, 3, 8, 8, 5, 8, 3, 1, 4, 1, 3, 2, 7, 9, 9, 9, 9, 9, 9, 3, 3, 6, 8, 1, 9, 7, 1, 6, 2, 6, 2, 1, 2, 9, 3, 7, 3, 4, 7, 9, 6, 8, 4, 7, 1, 7, 7, 3, 3, 0, 7, 6, 9, 8, 2, 0, 1, 5, 9, 9, 2, 1, 4, 2, 0, 0, 4, 0, 7, 8, 4, 9, 0, 8, 6, 5, 9, 2, 4, 8, 1, 7, 8, 7, 3, 9, 5, 5
Offset: 0

Views

Author

Clark Kimberling, Dec 18 2011

Keywords

Comments

For many choices of u and v, there is just one value of x satisfying u*x+v=e^(-x). Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 2.... A202322
1.... 3.... A202323
2.... 2.... A202353
2.... e.... A202354
1... -1.... A202355
1.... 0.... A030178
2.... 0.... A202356
e.... 0.... A202357
3.... 0.... A202392
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A202322, take f(x,u,v)=x+2-e^(-x) and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

Examples

			x=-0.442854401002388583141327999999336819716262...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    (* Program 1:  A202322 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.45, -.44}, WorkingPrecision -> 110]
RealDigits[r]  (* A202322 *)
(* Program 2: implicit surface of u*x+v=e^(-x) *)
f[{x_, u_, v_}] := u*x + v - E^-x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 1, 3}, {u, 1, 3}];
ListPlot3D[Flatten[t, 1]] (* for A202322 *)
RealDigits[ ProductLog[E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)
  • PARI
    lambertw(exp(2)) - 2 \\ G. C. Greubel, Jun 10 2017

Formula

x(u,v) = W(e^(v/u)/u) - v/u, where W = ProductLog = LambertW. - Jean-François Alcover, Feb 14 2013
Equals A226571 - 2 = LambertW(exp(2))-2. - Vaclav Kotesovec, Jan 09 2014

Extensions

Digits from a(84) on corrected by Jean-François Alcover, Feb 14 2013

A202617 E.g.f. satisfies: A(x) = exp( x*(1 + A(x)^2)/2 ).

Original entry on oeis.org

1, 1, 3, 19, 185, 2441, 40747, 823691, 19564785, 534145105, 16482667091, 567343245635, 21552042260905, 895664877901145, 40422799315249275, 1968883362773653051, 102942561775293158369, 5750760587905912310177, 341848844954020959953059, 21545207157567497255044979
Offset: 0

Views

Author

Paul D. Hanna, Dec 21 2011

Keywords

Comments

Compare to e.g.f. F(x) of A007889: F(x) = exp(x*(1 + F(x))/2), where A007889(n) = number of intransitive (or alternating) trees: vertices are [0,n] and for no i
Related sequence: A058014(n) = number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 185*x^4/4! + 2441*x^5/5! +...
where log(A(x)) = x*(1 + A(x)^2)/2 and
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + 23616*x^6/6! +...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 576*x^4/4! + 7872*x^5/5! + 134656*x^6/6! +...

Links

Crossrefs

Cf. A007889, A058014, A214225, A138860, A202357.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[-ProductLog[-E^x*x]/x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    a(n)=local(A=1+x); for(i=0, n, A=exp(x*(1+A^2)/2 +x*O(x^n))); n!*polcoeff(A, n)
  • PARI
    /* Coefficients of A(x)^p are given by: */
{a(n, p=1)=(1/2^n)*sum(k=0, n, binomial(n, k)*p*(2*k+p)^(n-1))}
  • PARI
    a(n)=n!*polcoeff(exp(sum(k=1,n,k^(k-1)*cosh(k*x +x*O(x^n))*x^k/k!) +x*O(x^n)),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012

Formula

E.g.f. A(x) equals the formal inverse of function 2*log(x)/(1+x^2).
E.g.f.: exp( Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n! ). - Paul D. Hanna, Nov 20 2012
E.g.f.: exp(G(x)) where G(x) = x/(1 - tanh(G(x))) is the e.g.f. of A214225. - Paul D. Hanna, Nov 20 2012
E.g.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the e.g.f. of A058014.
a(n) = (1/2^n)*Sum_{k=0..n} C(n,k)*(2*k+1)^(n-1).
Powers of e.g.f.:
If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(2*k+p)^(n-1).
a(n) ~ sqrt(1+c) * n^(n-1) / (2 * exp(n) * c^(n+1/2)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - Vaclav Kotesovec, Jan 10 2014
E.g.f.: sqrt(-LambertW(-x*exp(x))/x). - Vaclav Kotesovec, Jan 10 2014

A141251 Decimal expansion of the number c satisfying c*log(c)=1+c.

Original entry on oeis.org

3, 5, 9, 1, 1, 2, 1, 4, 7, 6, 6, 6, 8, 6, 2, 2, 1, 3, 6, 6, 4, 9, 2, 2, 2, 9, 2, 5, 7, 4, 1, 6, 3, 4, 8, 4, 2, 1, 0, 3, 0, 7, 5, 4, 0, 1, 5, 9, 2, 7, 8, 6, 9, 1, 9, 0, 4, 5, 2, 9, 8, 7, 3, 1, 9, 9, 2, 2, 6, 5, 4, 9, 8, 4, 4, 0, 3, 1, 6, 3, 7, 6, 6, 0, 2, 3, 6, 4, 1, 7, 7, 4, 6, 5, 2, 4, 5, 7, 1
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Sep 04 2008

Keywords

Comments

The iteration on c = c*log(c)- 1 does not converge from above or below. However, iteration on c = exp(1+1/c) converges quickly from above and below, including negative values. - Richard R. Forberg, Dec 28 2013
c is the solution to the equation 2 = Integral_{t=1...c} log(t) dt. - Colin Linzer, Nov 12 2024

Examples

			3.59112147666862213664922292574163484210307540159278691904529873...

Links

Crossrefs

Cf. A141252.

Programs

Formula

Equals exp(LambertW(1/e)+1).
Equals 1/A202357. - Hugo Pfoertner, Nov 12 2024

A215077 Binomial convolution of sum of consecutive powers.

Original entry on oeis.org

0, 1, 7, 66, 852, 14020, 280472, 6609232, 179317056, 5505532992, 188717617280, 7143999854464, 296013377405440, 13325516967972352, 647610246703508480, 33794224057227356160, 1884620857353101983744, 111857608180484932648960, 7040178644779119413723136, 468349192560992552808841216, 32836927387372039917034405888
Offset: 0

Views

Author

Olivier Gérard, Aug 02 2012

Keywords

Comments

a(0) could alternatively be defined as 1 from the formula or the convention for 0^0.
This sum is remarkable for its three different decompositions involving powers and binomials (see formulas and cross-refs).

Links

Crossrefs

Row sums of A215078, A215079, A215080.
See also A215083 and A215084.

Programs

  • Mathematica
    Table[Sum[Sum[j^n*Binomial[n, k], {j, 1, k}], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n)=sum(k=0,n, binomial(n,k)*sum(j=1,k, j^n)) \\ Charles R Greathouse IV, Jul 31 2016
  • PARI
    a(n)=my(P=sumformal('x^n)); sum(k=0,n, binomial(n,k)*subst(P,'x,k)) \\ Charles R Greathouse IV, Jul 31 2016

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*Sum_{j=1..k} j^n;
a(n) = Sum_{k=0..n} binomial(n,k)*H_k^{-n}, where H_k^(-n) = k-th harmonic number of order -n;
a(n) = Sum_{k=0..n} k^n * Sum_{j=0..n-k} binomial(n,n-k-j);
a(n) = Sum_{k=0..n} k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1);
a(n) = Sum_{k=0..n} Sum_{j=0..k} (k-j)^n * binomial(n,j);
a(n) = Sum_{k=0..n} Sum_{j=0..n} (n-j)^n * binomial(n,n+k-j);
and the equivalent formulas obtained by symmetries of the binomial and the hypergeometric function as well as treating the zeroth term separately.
a(n) ~ n^n / (sqrt(1+r) * (1-r) * exp(n) * r^n), where r = A202357 = LambertW(exp(-1)). - Vaclav Kotesovec, Jun 10 2019

A202355 Decimal expansion of the number x satisfying x-1=exp(-x).

Original entry on oeis.org

1, 2, 7, 8, 4, 6, 4, 5, 4, 2, 7, 6, 1, 0, 7, 3, 7, 9, 5, 1, 0, 9, 3, 5, 8, 7, 3, 9, 0, 2, 2, 9, 8, 0, 1, 5, 5, 4, 3, 9, 4, 7, 7, 4, 8, 8, 6, 1, 9, 7, 4, 5, 7, 6, 5, 4, 5, 3, 1, 7, 8, 1, 0, 5, 5, 3, 5, 0, 2, 9, 3, 7, 5, 4, 5, 9, 9, 4, 9, 8, 9, 8, 1, 9, 2, 0, 4, 9, 8, 4, 2, 8, 1, 1, 2, 9, 9, 4, 2
Offset: 1

Views

Author

Clark Kimberling, Dec 18 2011

Keywords

Comments

See A202322 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x=1.2784645427610737951093587390229801554394...

Links

Crossrefs

Cf. A202322.

Programs

  • Mathematica
    u = 1; v = -1;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r] (* A202355 *)
(* other program *)
RealDigits[1 + ProductLog[1/E], 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)
  • PARI
    1 + lambertw(exp(-1)) \\ G. C. Greubel, Jun 10 2017

Formula

Equals A202357 + 1. - Vaclav Kotesovec, Jan 31 2015

A202495 Decimal expansion of x satisfying x = e^(-2*Pi*x).

Original entry on oeis.org

2, 3, 2, 3, 1, 3, 3, 8, 2, 5, 5, 5, 1, 8, 1, 6, 2, 2, 8, 9, 5, 5, 2, 5, 4, 6, 6, 8, 0, 9, 0, 5, 4, 6, 9, 9, 6, 0, 0, 6, 5, 5, 4, 0, 3, 7, 2, 9, 1, 0, 6, 2, 4, 0, 8, 2, 6, 5, 4, 5, 6, 7, 1, 7, 8, 1, 0, 2, 2, 7, 8, 1, 9, 9, 3, 8, 2, 6, 8, 1, 7, 5, 3, 4, 2, 0, 8, 9, 8, 2, 1, 8, 5, 6, 9, 6, 8, 3, 6
Offset: 0

Views

Author

Clark Kimberling, Dec 20 2011

Keywords

Comments

See A202348 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x=0.232313382555181622895525466809054699600655...

Links

Crossrefs

Cf. A000796, A001113, A202348.

Programs

  • Maple
    evalf(LambertW(2*Pi)/(2*Pi), 145);  # Alois P. Heinz, Feb 26 2020
  • Mathematica
    u = -2*Pi; v = 0;
f[x_] := x; g[x_] := E^(u*x + v)
Plot[{f[x], g[x]}, {x, 0, .5}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .2, .3}, WorkingPrecision -> 110]
RealDigits[r]    (* A202357 *)
RealDigits[ ProductLog[2*Pi]/(2*Pi), 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

Formula

Equals LambertW(2*Pi)/(2*Pi). - Alois P. Heinz, Feb 26 2020

A253545 Decimal expansion of r = 0.527697..., a boundary ratio separating catenoid and Goldschmidt solutions in the minimal surface of revolution problem.

Original entry on oeis.org

5, 2, 7, 6, 9, 7, 3, 9, 6, 9, 6, 2, 5, 7, 1, 5, 2, 8, 5, 7, 2, 4, 2, 3, 3, 4, 3, 3, 6, 3, 1, 8, 0, 5, 7, 7, 9, 6, 8, 8, 5, 3, 7, 9, 0, 6, 3, 1, 4, 1, 9, 5, 4, 1, 7, 2, 2, 2, 7, 5, 1, 5, 9, 5, 0, 1, 6, 2, 0, 7, 6, 8, 3, 2, 4, 5, 1, 9, 8, 8, 4, 4, 6, 6, 8, 4, 5, 2, 9, 3, 6, 0, 0, 5, 4, 7, 5, 3, 0, 3, 5, 1, 4, 1, 5
Offset: 0

Views

Author

Jean-François Alcover, Apr 21 2015

Keywords

Comments

Consider two circular frames each of diameter D and with a separation of d.
If d/D < r = 0.527697..., then a catenoid gives the absolute minimum area.
If r < d/D < L = 0.66274... (Laplace limit), there are 3 minimal surfaces of revolution passing through the frames: 2 catenoids and the so-called Goldschmidt discontinuous solution consisting of the 2 disks.
If d/D > L, there remains only the Goldschmidt solution.

Examples

			0.5276973969625715285724233433631805779688537906314195417222751595...

Links

Crossrefs

Cf. A033259 (Laplace limit), A202357.

Programs

  • Mathematica
    digits = 105; u0 = u /. FindRoot[u*Sqrt[u^2-1] + ArcCosh[u] - u^2 == 0, {u, 6/5}, WorkingPrecision -> digits+5];  r = ArcCosh[u0]/u0; RealDigits[r, 10, digits] // First

Formula

arccosh(u)/u, where u = 1.21136... is solution to u*sqrt(u^2-1) + arccosh(u) - u^2 = 0.
Solution of 2*cosh((x^2+1)/2) = x+1/x. - Robert FERREOL, Feb 07 2019
Equals sqrt(A202357). - Hugo Pfoertner, Dec 21 2024

A186448 E.g.f. A(x) satisfies 2*A(x) = x*(1 + A(x) + exp(A(x))).

Original entry on oeis.org

1, 2, 10, 88, 1144, 19856, 432464, 11348352, 348715392, 12286859008, 488470565632, 21633197775872, 1056315874429952, 56382210082129920, 3266205054434912256, 204097766901573320704, 13684668496370094407680
Offset: 1

Views

Author

Vladimir Kruchinin, Feb 22 2011

Keywords

Comments

A(x) is e.g.f. for a(n)/(2^(n-1)/n). - Vaclav Kotesovec, Jan 26 2014

Links

Crossrefs

Cf. A202357.

Programs

  • Mathematica
    Table[(Rest[CoefficientList[InverseSeries[Series[(2*x)/(1+E^x+x),{x,0,20}],x],x]*Range[0,20]!])[[n]] * (2^(n-1)/n),{n,1,20}] (* Vaclav Kotesovec, Jan 26 2014 *)
Table[n!/(2*n+2)*Sum[Binomial[n+1,m]*Sum[m^(i)/i!*Binomial[n-m+1,n-i],{i,0,n}],{m,1,n}]+(n+1)^(n-1)/2+n!/2,{n,0,20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Jan 26 2014 *)
  • Maxima
    a(n):=n!/(2*n+2)*sum(binomial(n+1,m)*sum(m^(i)/i!*binomial(n-m+1,n-i),i,0,n),m,1,n)+(n+1)^(n-1)/2+(n)!/2;

Formula

a(n+1) = (n!/(2*n+2)) * (Sum_{m=1..n} binomial(n+1, m) * (Sum_{i=0..n} (m^i/i!) * binomial(n-m+1, n-i))) + (n+1)^(n-1)/2 + n!/2.
a(n) ~ n^(n-2) * (1+c)^(n+1) / (2 * c^n * exp(n)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - Vaclav Kotesovec, Jan 26 2014
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.