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.

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

A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

Original entry on oeis.org

1, 2, 4, 14, 106, 1732, 66634, 5745700, 1058905642, 461715853196, 461918527950694, 989913403174541980, 5009399946447021173140, 60070720443204091719085184, 1548154498059133199618813305334, 92346622775540905956057053976278584
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2009

Keywords

Comments

Row sums of A219206.

Links

Crossrefs

Cf. A000169, A014062, A167009, A167010, A184731, A219206, A220359, A362288.

Programs

  • Haskell
    a167008 = sum . a219206_row  -- Reinhard Zumkeller, Feb 27 2015
  • Magma
    [(&+[Binomial(n,j)^j: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022
  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n, k]^k, {k,0,n}], {n,20}]}]
(* Program for numerical value of the limit a(n)^(1/n^2) *) (1-r)^(-r/2)/.FindRoot[(1-r)^(2*r-1)==r^(2*r),{r,1/2},WorkingPrecision->100] (* Vaclav Kotesovec, Dec 12 2012 *)
Total/@Table[Binomial[n,k]^k,{n,0,20},{k,0,n}] (* Harvey P. Dale, Oct 19 2021 *)
  • PARI
    a(n)=sum(k=0,n,binomial(n,k)^k)
  • SageMath
    [sum(binomial(n,j)^j for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Formula

Limit_{n->oo} 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, Dec 12 2012

A219207 Triangle, read by rows, where T(n,k) = binomial(n,k)^(k+1) for n>=0, k=0..n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 27, 1, 1, 16, 216, 256, 1, 1, 25, 1000, 10000, 3125, 1, 1, 36, 3375, 160000, 759375, 46656, 1, 1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1, 1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1, 1, 81, 46656
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2012

Keywords

Comments

Maximal term in row n is asymptotically in position k = r*n, where r = A220359 = 0.70350607643... is a root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Nov 15 2012

Examples

			Triangle of coefficients C(n,k)^(k+1) begins:
1;
1, 1;
1, 4, 1;
1, 9, 27, 1;
1, 16, 216, 256, 1;
1, 25, 1000, 10000, 3125, 1;
1, 36, 3375, 160000, 759375, 46656, 1;
1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1;
1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1; ...
MATRIX INVERSE.
The matrix inverse starts
1;
-1,1;
3,-4,1;
-73,99,-27,1;
18055,-24496,6696,-256,1;
-55694851,75563975,-20656000,790000,-3125,1; - _R. J. Mathar_, Mar 22 2013

Links

Crossrefs

Cf. A184731, A219206, A184730.

Programs

  • Mathematica
    Table[Binomial[n,k]^(k+1),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 15 2016 *)
  • PARI
    {T(n,k)=binomial(n,k)^(k+1)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

Row sums equal A184731.

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

A295610 a(n) = Sum_{k=0..n} (n!/(n - k)!)^k.

Original entry on oeis.org

1, 2, 7, 256, 345749, 25090776406, 139507578065088907, 82622801516492599819822772, 6985137485409222182920705065038896201, 109110989095384931538566720095053550173384985449034, 395940975233113726268241745444050219538058574725338743701918216111
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2017

Keywords

Links

Crossrefs

Cf. A000522, A006040, A036740, A217284, A219206.

Programs

  • Mathematica
    Table[Sum[(n!/(n - k)!)^k, {k, 0, n}], {n, 0, 10}]
Table[Sum[(Gamma[n + 1]/Gamma[k + 1])^(n - k), {k, 0, n}], {n, 0, 10}]
Table[Sum[(Binomial[n, k] k!)^k, {k, 0, n}], {n, 0, 10}]
  • PARI
    a(n) = sum(k=0, n, (n!/(n - k)!)^k); \\ Michel Marcus, Nov 25 2017

Formula

a(n) = Sum_{k=0..n} A219206(n,k)*A036740(k).
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n^2 + n/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Nov 25 2017
Showing 1-5 of 5 results.

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