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.

A201556 G.f.: exp( Sum_{n>=1} C(2*n^2,n^2) * x^n/n ).

Original entry on oeis.org

1, 2, 37, 16278, 150303194, 25282422428664, 73752140616074524401, 3639659041645240391812731402, 2993893262520330875797362908273443346, 40656420461436928818704580402413441308206341488, 9054851465691640957562090101797213977192016103053025996396
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2011

Keywords

Comments

Self-convolution of A213402.
Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108): C(x)^2 = exp( Sum_{n>=1} binomial(2*n,n) * x^n/n ).

Examples

			G.f.: A(x) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 +...
where
log(A(x)) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 +...+ C(2*n^2,n^2)*x^n/n +...

Links

Crossrefs

Cf. A213402, A201555, A000984, A200002, A213409.

Programs

  • Mathematica
    nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k^2, k^2]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m^2,m^2)*x^m/m)+x*O(x^n)),n)}
  • PARI
    {a(n)=if(n==0,1,(1/n)*sum(k=1,n,binomial(2*k^2,k^2)*a(n-k)))}

Formula

a(n) = (1/n) * Sum_{k=1..n} C(2*k^2,k^2) * a(n-k) for n>0 with a(0)=1.
a(n) ~ 4^(n^2) / (sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014

A245245 a(n) = C(3*n^2, n^2).

Original entry on oeis.org

1, 3, 495, 4686825, 2254848913647, 52588547141148893628, 58152371703925106867047535565, 3012179439602547459232394950891834843500, 7255167425905233148164780983569428433097979870294255, 808718755397067598640202627155266231883064669446721506930287016061
Offset: 0

Views

Author

Paul D. Hanna, Jul 15 2014

Keywords

Comments

a(n) = A005809(n^2).

Links

Crossrefs

Cf. A245243, A201555, A005809, A213409 (exp).

Programs

  • Mathematica
    Table[Binomial[3*n^2, n^2],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)
  • PARI
    {a(n)=binomial(3*n^2, n^2)}
for(n=0,15,print1(a(n),", "))

Formula

Logarithmic derivative of A213409 (ignoring initial term a(0) of this sequence).
a(n) ~ 3^(3*n^2+1/2) / (sqrt(Pi) * n * 2^(2*n^2+1)). - Vaclav Kotesovec, Jul 15 2014

A213410 G.f.: exp( Sum_{n>=1} binomial(3*n,n)^n/3^n * x^n/n ).

Original entry on oeis.org

1, 1, 13, 7330, 185307558, 201002187396362, 9357300769149011773697, 18775362849239140086719414696830, 1631039199744298058694966065590003308698494, 6159916689356522044764167426829149420348399496664634288
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2012

Keywords

Comments

Compare to the g.f. G(x) = 1 + x*G(x)^3 of A001764: G(x) = exp( Sum_{n>=1} binomial(3*n,n)/3 * x^n/n ).

Examples

			G.f.: A(x) = 1 + x + 13*x^2 + 7330*x^3 + 185307558*x^4 + 201002187396362*x^5 +...
where
log(A(x)) = x + 5^2*x^2/2 + 28^3*x^3/3 + 165^4*x^4/4 + 1001^5*x^5/5 + 6188^6*x^6/6 + 38760^7*x^7/7 +...+ A025174(n)^n*x^n/n +...

Crossrefs

Cf A213409, A001764, A025174, A200002.

Programs

  • Mathematica
    nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*k,k]^k/3^k * b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, binomial(3*m, m)^m/3^m*x^m/m)+x*O(x^n)), n)}
for(n=0,15,print1(a(n),", "))

Formula

a(n) = (1/n) * Sum_{k=1..n} binomial(3*k,k)^k/3^k * a(n-k) for n>0 with a(0)=1.
Showing 1-3 of 3 results.

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