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 1-3 of 3 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

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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

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 15, 1, 1, 4, 70, 220, 1, 1, 5, 210, 5005, 4845, 1, 1, 6, 495, 48620, 735471, 142506, 1, 1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1, 1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1, 1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2013

Keywords

Comments

Central coefficients are A201555(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

Examples

			The triangle of coefficients C(n*k, k^2), n>=k, k=0..n, begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1;
1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1;
1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1; ...

Links

Crossrefs

Cf. A228808 (row sums), A228833 (antidiagonal sums), A135860 (diagonal), A201555 (central terms).
Cf. A229052.
Cf. related triangles: A228904 (exp), A209330, A226234, A228836.

Programs

  • PARI
    {T(n, k)=binomial(n*k, k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A228809 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) ).

Original entry on oeis.org

1, 2, 4, 12, 94, 2195, 158904, 31681195, 13904396167, 15305894726347, 44888344014554903, 288228807835914177564, 4270880356112396772814732, 169380654509201278629725097906, 15394658527137259981745081997280638, 3042352591056504014301304188228238554499
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2013

Keywords

Comments

Logarithmic derivative equals A228808.
Equals row sums of triangle A228904.

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 +...
where
log(A(x)) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 + 927100*x^6/6 +...+ A228808(n)*x^n/n +...

Links

Crossrefs

Cf. A228808, A207135, A228904.
Cf. variants: A167006, A206848.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, x^m/m*sum(k=0, m, binomial(m*k, k^2)))+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
Showing 1-3 of 3 results.

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