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

A219206 Triangle, read by rows, where T(n,k) = binomial(n,k)^k for n>=0, k=0..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 9, 1, 1, 4, 36, 64, 1, 1, 5, 100, 1000, 625, 1, 1, 6, 225, 8000, 50625, 7776, 1, 1, 7, 441, 42875, 1500625, 4084101, 117649, 1, 1, 8, 784, 175616, 24010000, 550731776, 481890304, 2097152, 1, 1, 9, 1296, 592704, 252047376, 31757969376, 351298031616, 78364164096, 43046721, 1
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 begins:
  1;
  1, 1;
  1, 2,   1;
  1, 3,   9,      1;
  1, 4,  36,     64,        1;
  1, 5, 100,   1000,      625,         1;
  1, 6, 225,   8000,    50625,      7776,         1;
  1, 7, 441,  42875,  1500625,   4084101,    117649,       1;
  1, 8, 784, 175616, 24010000, 550731776, 481890304, 2097152,  1;
  ...

Links

Crossrefs

Cf. A167008 (row sums).
Cf. A002262, A007318, A219207.

Programs

  • Haskell
    a219206 n k = a219206_tabl !! n !! k
a219206_row n = a219206_tabl !! n
a219206_tabl = zipWith (zipWith (^)) a007318_tabl a002262_tabl
-- Reinhard Zumkeller, Feb 27 2015
  • PARI
    {T(n,k)=binomial(n,k)^k}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

Row sums equal A167008.

A228899 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 12, 1, 1, 10, 71, 76, 1, 1, 15, 281, 2153, 701, 1, 1, 21, 861, 29166, 129509, 8477, 1, 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1, 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1, 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2013

Keywords

Comments

Note that the following g.f. does NOT yield an integer triangle: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^k * y^k ).

Examples

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 6, 12, 1;
1, 10, 71, 76, 1;
1, 15, 281, 2153, 701, 1;
1, 21, 861, 29166, 129509, 8477, 1;
1, 28, 2212, 244725, 7664343, 12391414, 126126, 1;
1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1;
1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1; ...
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+12*y^2+y^3)*x^3 + (1+10*y+71*y^2+76*y^3+y^4)*x^4 + (1+15*y+281*y^2+2153*y^3+701*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x
+ (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^2*y + 3^3*y^2 + y^3)*x^3/3
+ (1+ 4^2*y + 6^3*y^2 + 4^4*y^3 + x^4)*x^4/4
+ (1+ 5^2*y + 10^3*y^2 + 10^4*y^3 + 5^5*y^4 + y^5)*x^5/5
+ (1+ 6^2*y + 15^3*y^2 + 20^4*y^3 + 15^5*y^4 + 6^6*y^5 + y^6)*x^6/6 +...
in which the coefficients form A219207(n,k) = binomial(n, k)^(k+1).

Crossrefs

Cf. A184730 (row sums), A181070 (antidiagonal sums), A060946 (diagonal).
Cf. related triangles: A219207, A209424, A228904.

Programs

  • PARI
    {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m, j)^(j+1)*y^j))+x*O(x^n)), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
Showing 1-3 of 3 results.

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