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.

A226750 Decimal expansion of the number x other than -3 defined by x*e^x = -3/e^3.

Original entry on oeis.org

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

Views

Author

José María Grau Ribas, Jun 16 2013

Keywords

Examples

			-0.178560627877921106596808669705514804654118255926885907720142306058785...

Links

Crossrefs

Cf. A106533, A217913, A221214, A274712, A274713.

Programs

  • Mathematica
    RealDigits[N[ProductLog[-3/e^3], 105]][[1]]
  • PARI
    solve(x=0,1,3/exp(3)-x*exp(-x)) \\ Charles R Greathouse IV, Nov 19 2013

Extensions

Term a(104) corrected by G. C. Greubel, Nov 16 2017

A242449 a(n) = Sum_{k=0..n} C(n,k) * (2*k+1)^(2*n+1).

Original entry on oeis.org

1, 28, 3612, 1064480, 560632400, 462479403072, 550095467201728, 891290348282967040, 1887146395301619304704, 5058811707344107766328320, 16746136671945501439084657664, 67088193422344140016282100785152, 319900900946743851959321101768511488
Offset: 0

Views

Author

Vaclav Kotesovec, May 14 2014

Keywords

Comments

Generally, for p>=1, a(n) = Sum_{k=0..n} C(n,k) * (p*k+1)^(p*n+1) is asymptotic to n^(p*n+1) * p^(p*n+1) * r^(p*n+3/2+1/p) / (sqrt(p+r-p*r) * exp(p*n) * (1-r)^(n+1/p)), where r = p/(p+LambertW(p*exp(-p))).

Links

Crossrefs

Cf. A242373, A220955, A242446, A072034, A007820, A221214, A213193, A195242.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*(2*k+1)^(2*n+1),{k,0,n}],{n,0,20}]
  • PARI
    for(n=0,30, print1(sum(k=0,n, binomial(n,k)*(2*k+1)^(2*n+1)), ", ")) \\ G. C. Greubel, Nov 16 2017

Formula

a(n) ~ n^(2*n+1) * 2^(2*n+1) * r^(2*n+2) / (sqrt(2-r) * exp(2*n) * (1-r)^(n+1/2)), where r = 2/(2+LambertW(2*exp(-2))) = 0.901829091937052...

A220955 O.g.f.: Sum_{n>=0} (2*n+1)^(2*n+1) * exp(-(2*n+1)^2*x) * x^n / n!.

Original entry on oeis.org

1, 26, 1320, 99288, 9901920, 1230768704, 183260197120, 31800433551744, 6301891570411008, 1404224096732154880, 347532097449969496064, 94584986134590717358080, 28076463606243146379018240, 9027122730610037995425792000, 3125219575155651450096795648000
Offset: 0

Views

Author

Paul D. Hanna, Feb 27 2013

Keywords

Comments

From Vaclav Kotesovec, May 13 2014: (Start)
Generally, for p>1, a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (p*k+1)^(p*n+1) = Sum_{k=0..(p-1)*n+1} p^(n+k) * binomial(p*n+1,n+k) * stirling2(n+k,n).
a(n) ~ n^(n*p-n+1/2) * p^(2*p*n+2+1/p) / (sqrt(2*Pi*(1-r)) * exp((p-1)*n) * r^(n+1/p) * (p-r)^(n*p-n+1)), where r = -LambertW(-p*exp(-p)).
(End)

Examples

			O.g.f.: A(x) = 1 + 26*x + 1320*x^2 + 99288*x^3 + 9901920*x^4 +...
where A(x) = exp(-x) + 3^3*exp(-3^2*x)*x + 5^5*exp(-5^2*x)*x^2/2! + 7^7*exp(-7^2*x)*x^3/3! + 9^9*exp(-9^2*x)*x^4/4! + 11^11*exp(-11^2*x)*x^5/5! +...
is a power series in x with integer coefficients.

Links

Crossrefs

Cf. A222525, A221214, A213193, A217900, A007820, A242449.

Programs

  • Mathematica
    Table[1/n! * Sum[(-1)^(n-k)*Binomial[n,k] * (2*k+1)^(2*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[2*n+1,n+k]*2^(n+k)*StirlingS2[n+k,n],{k,0,n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
  • PARI
    {a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*exp(-(2*k+1)^2*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=(1/n!)*polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*x^k/(1+(2*k+1)^2*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k+1)^(2*n+1))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} (2*k+1)^(2*k+1) * x^k / (1 + (2*k+1)^2*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (2*k+1)^(2*n+1).
a(n) = Sum_{k=0..n+1} 2^(n+k) * binomial(2*n+1,n+k) * stirling2(n+k,n). - Vaclav Kotesovec, May 13 2014
a(n) ~ n^(n+1/2) * 2^(4*n+5/2) / (sqrt(2*Pi*(1-r)) * exp(n) * r^(n+1/2) * (2-r)^(n+1)), where r = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775 = -r) . - Vaclav Kotesovec, May 13 2014

A213193 O.g.f.: Sum_{n>=0} (4*n+1)^(4*n+1) * exp(-(4*n+1)^4*x) * x^n / n!.

Original entry on oeis.org

1, 3124, 191757120, 49208861869440, 33030777426968816640, 45829974166034718596428800, 114009204539207742166715857223680, 462192193445890293982679086838571270144, 2851153321165202191241172917762717987236478976
Offset: 0

Views

Author

Paul D. Hanna, Mar 01 2013

Keywords

Comments

From Vaclav Kotesovec, May 13 2014: (Start)
Generally, for p>1, a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (p*k+1)^(p*n+1) = Sum_{k=0..(p-1)*n+1} p^(n+k) * binomial(p*n+1,n+k) * stirling2(n+k,n).
a(n) ~ n^(n*p-n+1/2) * p^(2*p*n+2+1/p) / (sqrt(2*Pi*(1-r)) * exp((p-1)*n) * r^(n+1/p) * (p-r)^(n*p-n+1)), where r = -LambertW(-p*exp(-p)).
(End)

Examples

			O.g.f.: A(x) = 1 + 3124*x + 191757120*x^2 + 49208861869440*x^3 +...
where
A(x) = exp(-x) + 5^5*x*exp(-5^4*x) + 9^9*exp(-9^4*x)*x^2/2! + 13^13*exp(-13^4*x)*x^3/3! + 17^17*exp(-17^4*x)*x^4/4! + 21^21*exp(-21^4*x)*x^5/5! +...
is a power series in x with integer coefficients.

Crossrefs

Cf. A222525, A220955, A221214, A217900, A007820, A242449.

Programs

  • Mathematica
    Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(4*k+1)^(4*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[4*n+1,n+k]*4^(n+k)*StirlingS2[n+k,n],{k,0,3*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
  • PARI
    {a(n)=polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*exp(-(4*k+1)^4*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=(1/n!)*polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*x^k/(1+(4*k+1)^4*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(4*k+1)^(4*n+1))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} (4*k+1)^(4*k+1) * x^k / (1 + (4*k+1)^4*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (4*k+1)^(4*n+1).
a(n) ~ n^(3*n+1/2) * 2^(16*n+9/2) / (sqrt(2*Pi*(1-r)) * exp(3*n) * r^(n+1/4) * (4-r)^(3*n+1)), where r = -LambertW(-4*exp(-4)) = 0.0793096051271136564391... . - Vaclav Kotesovec, May 13 2014
Showing 1-4 of 4 results.

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