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.

A217615 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 15, 29, 49, 95, 187, 345, 659, 1289, 2465, 4739, 9237, 17911, 34715, 67705, 132063, 257477, 503309, 984983, 1927895, 3778017, 7411237, 14544967, 28565661, 56144615, 110406527, 217225533, 427636561, 842256047, 1659600955, 3271579689, 6451913519
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2012

Keywords

Comments

Radius of convergence of g.f. is r = 1/2.
More generally, given
A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
then A(x) = 1/sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t > 0.
a(n) is the number of (2k-1)-element subsets of {1, 2, ..., n+1} whose k-th smallest (i.e., k-th largest) element equals 2k-1. - Darij Grinberg, Oct 09 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 15*x^6 + 29*x^7 + 49*x^8 + ...
where the g.f. equals the series:
A(x) = 1 +
  x*((1-x) + x) +
  x^2*((1-x)^2 + 2^2*x*(1-x) + x^2) +
  x^3*((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3) +
  x^4*((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4) +
  x^5*((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5) + ...

Links

Crossrefs

Cf. A217616, A217617, A217461, A217464, A216604, A217661.

Programs

  • Maple
    a := n -> `if`(n < 4, [1, 1, 1, 3][n+1], hypergeom([1/2, (1-n)/3, (2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27)):
seq(simplify(a(n)), n=0..36); # Peter Luschny, Oct 09 2019
  • Mathematica
    CoefficientList[Series[1/Sqrt[(1-x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2013 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k) + x*O(x^n))), n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    a(n)={sum(k=0, n\2, binomial(2*k, k) * binomial(n-2*k, k))} \\ Andrew Howroyd, Oct 09 2019

Formula

G.f.: A(x) = 1 / sqrt( (1 - x + 2*x^2)^2 - 4*x^2 ).
G.f.: A(x) = 1 / sqrt( (1-x)*(1-2*x)*(1+x+2*x^2) ).
G.f. satisfies: A(x) = (1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)) / (1-x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.
a(n) ~ 2^n/sqrt(Pi*n). - Vaclav Kotesovec, Sep 16 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k, k) * binomial(n-2*k, k). - Darij Grinberg, Oct 09 2019
a(n) = hypergeom([1/2,(1-n)/3,(2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27) for n >= 4. - Peter Luschny, Oct 09 2019

A217665 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-3*x)^k.

Original entry on oeis.org

1, 1, 2, 8, 32, 122, 462, 1758, 6718, 25750, 98956, 381196, 1471678, 5693146, 22064296, 85655812, 333035302, 1296684130, 5055195944, 19731318068, 77098776372, 301561031472, 1180608808044, 4626045139116, 18140934734434, 71191952221114, 279576978531644
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2012

Keywords

Comments

Radius of convergence of g.f. A(x) is |x| < 1/4.
More generally, given
A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 32*x^4 + 122*x^5 + 462*x^6 + 1758*x^7 +...
where the g.f. equals the series:
A(x) = 1 +
x*(1 + x/(1-3*x)) +
x^2*(1 + 2^2*x/(1-3*x) + x^2/(1-3*x)^2) +
x^3*(1 + 3^2*x/(1-3*x) + 3^2*x^2/(1-3*x)^2 + x^3/(1-3*x)^3) +
x^4*(1 + 4^2*x/(1-3*x) + 6^2*x^2/(1-3*x)^2 + 4^2*x^3/(1-3*x)^3 + x^4/(1-3*x)^4) +
x^5*(1 + 5^2*x/(1-3*x) + 10^2*x^2/(1-3*x)^2 + 10^2*x^3/(1-3*x)^3 + 5^2*x^4/(1-3*x)^4 + x^5/(1-3*x)^5) +...

Links

Crossrefs

Cf. A217661, A217664, A217666.

Programs

  • Mathematica
    CoefficientList[Series[(1-3*x)/Sqrt[(1-4*x)*(1 - 4*x + 4*x^2 - 4*x^3)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 17 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-3*x +x*O(x^n))^k )), n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: (1-3*x) / sqrt(1 - 8*x + 20*x^2 - 20*x^3 + 16*x^4).
G.f.: (1-3*x) / sqrt( (1-4*x)*(1 - 4*x + 4*x^2 - 4*x^3) ).
a(n) ~ 4^n / (sqrt(3*Pi*n)). - Vaclav Kotesovec, Feb 17 2014

A217664 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-2*x)^k.

Original entry on oeis.org

1, 1, 2, 7, 23, 72, 227, 725, 2332, 7537, 24465, 79720, 260633, 854539, 2808768, 9252315, 30536925, 100959558, 334301159, 1108483583, 3680134756, 12231824111, 40697552035, 135536687436, 451776392011, 1507088458381, 5031254413136, 16807872970501, 56185887793379
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2012

Keywords

Comments

Radius of convergence of g.f. A(x) is |x| < 0.29392962790...
More generally, given
A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 23*x^4 + 72*x^5 + 227*x^6 + 725*x^7 +...
where the g.f. equals the series:
A(x) = 1 +
x*(1 + x/(1-2*x)) +
x^2*(1 + 2^2*x/(1-2*x) + x^2/(1-2*x)^2) +
x^3*(1 + 3^2*x/(1-2*x) + 3^2*x^2/(1-2*x)^2 + x^3/(1-2*x)^3) +
x^4*(1 + 4^2*x/(1-2*x) + 6^2*x^2/(1-2*x)^2 + 4^2*x^3/(1-2*x)^3 + x^4/(1-2*x)^4) +
x^5*(1 + 5^2*x/(1-2*x) + 10^2*x^2/(1-2*x)^2 + 10^2*x^3/(1-2*x)^3 + 5^2*x^4/(1-2*x)^4 + x^5/(1-2*x)^5) +...

Crossrefs

Cf. A217661, A217665, A217666.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-2*x +x*O(x^n))^k )), n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: (1-2*x) / sqrt(1 - 6*x + 11*x^2 - 10*x^3 + 9*x^4).

A217666 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-4*x)^k.

Original entry on oeis.org

1, 1, 2, 9, 43, 198, 903, 4121, 18840, 86255, 395397, 1814662, 8337729, 38350063, 176574336, 813785593, 3753980313, 17332179596, 80089232683, 370370470791, 1714045215632, 7938075605697, 36787429315319, 170592514889814, 791557946825363, 3674974608196665
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2012

Keywords

Comments

Radius of convergence of g.f. A(x) is |x| < 0.2116085881629750...
More generally, given
A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 43*x^4 + 198*x^5 + 903*x^6 + 4121*x^7 +...
where the g.f. equals the series:
A(x) = 1 +
x*(1 + x/(1-4*x)) +
x^2*(1 + 2^2*x/(1-4*x) + x^2/(1-4*x)^2) +
x^3*(1 + 3^2*x/(1-4*x) + 3^2*x^2/(1-4*x)^2 + x^3/(1-4*x)^3) +
x^4*(1 + 4^2*x/(1-4*x) + 6^2*x^2/(1-4*x)^2 + 4^2*x^3/(1-4*x)^3 + x^4/(1-4*x)^4) +
x^5*(1 + 5^2*x/(1-4*x) + 10^2*x^2/(1-4*x)^2 + 10^2*x^3/(1-4*x)^3 + 5^2*x^4/(1-4*x)^4 + x^5/(1-4*x)^5) +...

Crossrefs

Cf. A217661, A217664, A217665.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-4*x +x*O(x^n))^k )), n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: (1-4*x) / sqrt(1 - 10*x + 31*x^2 - 34*x^3 + 25*x^4).
Showing 1-4 of 4 results.

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