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.

A206153 G.f.: exp( Sum_{n>=1} A206154(n)*x^n/n ), where A206154(n) = Sum_{k=0..n} binomial(n,k)^(k+2).

Original entry on oeis.org

1, 2, 7, 48, 693, 26632, 2542514, 533442978, 278979307990, 343728261289376, 904762216681139381, 5771110378770242683658, 88742047516327429085056353, 2912737209806573079629325613400, 224604736339682169442980060945290802
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Logarithmic derivative yields A206154.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 + 14921404*x^6/6 +...+ A206154(n)*x^n/n +...

Crossrefs

Cf. A206154 (log), A184730, A206155, A206157, A206151.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(k+2))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))

A220359 Decimal expansion of the root of the equation (1-r)^(2*r-1) = r^(2*r).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Dec 12 2012

Keywords

Comments

Constant is associated with A167008, A219206 and A219207.

Examples

			0.70350607643066243...

Crossrefs

Cf. A167008, A219206, A219207, A228808, A184731, A206154, A206156, A206158, A237421, A295611.

Programs

  • Maple
    Digits:= 140:
v:= convert(fsolve( (1-r)^(2*r-1) = r^(2*r), r=1/2), string):
seq(parse(v[n+2]), n=0..120);  # Alois P. Heinz, Dec 12 2012
  • Mathematica
    RealDigits[r/.FindRoot[(1-r)^(2*r-1)==r^(2*r),{r,1/2}, WorkingPrecision->250], 10, 200][[1]]
  • PARI
    solve(x=.7,1,(1-x)^(2*x-1) - x^(2*x)) \\ Charles R Greathouse IV, Apr 25 2016

A206156 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).

Original entry on oeis.org

1, 2, 6, 92, 5410, 1400652, 2687407464, 18947436116184, 536104663173431874, 130559883231879141946580, 136031455187223511721647272376, 483565526783420050082035900177878504, 14487924180895151383693101563813954330590756
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A206155.

Examples

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 +...
where exponentiation yields A206155:
exp(L(x)) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 + 448468978*x^6 +...
Illustration of initial terms:
a(1) = 1^0 + 1^2 = 2;
a(2) = 1^0 + 2^2 + 1^4 = 6;
a(3) = 1^0 + 3^2 + 3^4 + 1^6 = 92;
a(4) = 1^0 + 4^2 + 6^4 + 4^6 + 1^8 = 5410;
a(5) = 1^0 + 5^2 + 10^4 + 10^6 + 5^8 + 1^10 = 1400652; ...

Crossrefs

Cf. A206155 (exp), A184731, A206154, A206158, A206152, A220359.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^(2*k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^(2*k))}
for(n=0,16,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Mar 03 2014

A206158 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).

Original entry on oeis.org

1, 2, 10, 272, 24226, 12053252, 40086916024, 429254371605824, 23527609330364490754, 10714627376371224032350052, 16964729291782419425708732425300, 109783535843179466164398767001178968704, 6782057095273243388704415924996348722446049600
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A206157.

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 +...
where exponentiation yields A206157:
exp(L(x)) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 + 6686021554*x^6 +...
Illustration of initial terms:
a(1) = 1^1 + 1^3 = 2;
a(2) = 1^1 + 2^3 + 1^5 = 10;
a(3) = 1^1 + 3^3 + 3^5 + 1^7 = 272;
a(4) = 1^1 + 4^3 + 6^5 + 4^7 + 1^9 = 24226;
a(5) = 1^1 + 5^3 + 10^5 + 10^7 + 5^9 + 1^11 = 12053252; ...

Crossrefs

Cf. A206157 (exp), A184731, A206154, A206156, A206152, A220359.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^(2*k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^(2*k+1))}
for(n=0,16,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Mar 03 2014
Showing 1-4 of 4 results.

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