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.

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A228808 a(n) = Sum_{k=0..n} binomial(n*k, k^2).

Original entry on oeis.org

1, 2, 4, 20, 296, 10067, 927100, 219541877, 110728186648, 137502766579907, 448577320868198789, 3169529341990169816462, 51243646781214826181569316, 2201837465728010770618930322223, 215520476721579201896200887266792583, 45634827026091489574547858030506357191920
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2013

Keywords

Comments

Ignoring initial term, equals the logarithmic derivative of A228809.
Equals row sums of triangle A228832.

Examples

			L.g.f.: L(x) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 +...
where
exp(L(x)) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 + 158904*x^6 + 31681195*x^7 +...+ A228809(n)*x^n +...

Links

Crossrefs

Cf. A228809, A207136, A167009, A228832.

Programs

  • Mathematica
    Table[Sum[Binomial[n*k, k^2],{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Sep 06 2013 *)
  • PARI
    a(n)=sum(k=0,n,binomial(n*k,k^2))
for(n=0,20,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Sep 06 2013

A323773 Decimal expansion of the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 27 2019

Keywords

Examples

			0.3663201503052830964087236563781171194011826607210994595491823160184...

Crossrefs

Cf. A220359, A228833, A237421, A245259, A323768, A323777, A323778.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[(1-2*r)^(4*r-1) * (1-r)^(1-2*r) == r^(2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A323778 Decimal expansion of the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 27 2019

Keywords

Examples

			0.36549849821985804457973687544629908832275880696346029501595516768211883674...

Crossrefs

Cf. A123165, A206851.
Cf. A220359, A237421, A245259, A323768, A323773, A323777.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[(1-r)^(2-2*r) * r^(2*r) == 1-2*r, {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A295611 a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^k.

Original entry on oeis.org

1, 0, 0, 6, -30, -280, 35070, -2508268, -47103462, 241470400824, -256752145545390, 128291714550379292, 2203924344437376054780, -37693423679943326954848176, 485163732930867224220253809178, 27101025121379607823580070619517816
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2017

Keywords

Links

Crossrefs

Cf. A167008, A167010, A219206.

Programs

  • Mathematica
    Table[Sum[(-1)^k Binomial[n, k]^k, {k, 0, n}], {n, 0, 15}]
Table[Sum[(-1)^k (n!/(k! (n - k)!))^k, {k, 0, n}], {n, 0, 15}]
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n,k)^k); \\ Michel Marcus, Nov 25 2017

Formula

a(n) = Sum_{k=0..n} (-1)^k*A219206(n,k).
Limit n->infinity |a(n)|^(1/n^2) = r^(r^2/(1-2*r)) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243096929661621777... is the real root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Nov 25 2017

A245260 Decimal expansion of the root of the equation r*log(r/(1-r))=1.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jul 15 2014

Keywords

Examples

			0.78218829428019990122029707592674478018190840396629951687...

Crossrefs

Cf. A072034, A220359, A237421, A245259.

Programs

  • Maple
    evalf(solve(r*log(r/(1-r))=1), 100)
  • Mathematica
    RealDigits[r/.FindRoot[r*Log[r/(1-r)]==1, {r, 3/4}, WorkingPrecision->250], 10, 200][[1]]
RealDigits[1/(1+LambertW[E^(-1)]), 10, 200][[1]]

Formula

Equals 1/(1+LambertW(exp(-1))).

A358496 a(n) = Sum_{k=0..n} binomial(binomial(n, k), k).

Original entry on oeis.org

1, 2, 3, 7, 24, 176, 2623, 79479, 5141566, 669156932, 178757299486, 104033138190939, 125893536876304530, 320091464865316176891, 1828276720220263211454403, 22393381352339181425954204921, 582288411818399885839904060337943, 34678571156322738984042119670750665153
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 19 2022

Keywords

Crossrefs

Cf. A167008, A220359, A357871, A358495.

Programs

  • Mathematica
    Table[Sum[Binomial[Binomial[n, k], k], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(binomial(n, k), k)); \\ Michel Marcus, Nov 19 2022

Formula

Limit_{n->infinity} a(n)^(1/n^2) = r^(r^2/(1-2*r)) = 1.533628065110458582..., where r = A220359 = 0.70350607643066243096929661621777... is the real root of the equation (1-r)^(2*r-1) = r^(2*r).
Previous Showing 11-16 of 16 results.

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