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 1-2 of 2 results.

A213409 G.f.: exp( Sum_{n>=1} binomial(3*n^2,n^2) * x^n/n ).

Original entry on oeis.org

1, 3, 252, 1563022, 563716946982, 10517711119760250261, 9692061982207456039533424586, 430311348543725825536505706371595438684, 906895928239445077568583988067142630846220290783969, 89857639488565787203362892584824012528872539028234934088960440084
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)^3 = exp( Sum_{n>=1} binomial(3*n,n) * x^n/n ).

Examples

			G.f.: A(x) = 1 + 3*x + 252*x^2 + 1563022*x^3 + 563716946982*x^4 +...
where
log(A(x)) = 3*x + 495*x^2/2 + 4686825*x^3/3 + 2254848913647*x^4/4 + 52588547141148893628*x^5/5 +...+ C(3*n^2,n^2)*x^n/n +...

Links

Crossrefs

Cf. A201556, A213410, A001764, A245245 (log).

Programs

  • Mathematica
    nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*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(3*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(3*k^2, k^2)*a(n-k)))}

Formula

a(n) = (1/n) * Sum_{k=1..n} binomial(3*k^2,k^2) * a(n-k) for n>0 with a(0)=1.
a(n) ~ sqrt(3) * (27/4)^(n^2) / (2*sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014

A245243 Triangle, read by rows, defined by T(n,k) = C(n^2 - k^2, n*k - k^2), for k=0..n, n>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 28, 10, 1, 1, 455, 495, 35, 1, 1, 10626, 54264, 8008, 126, 1, 1, 324632, 10518300, 4686825, 125970, 462, 1, 1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1, 1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2014

Keywords

Comments

Row sums equal A245242.
Central terms are A245245(n) = C(3*n^2, n^2).

Examples

			Triangle T(n,k) = C(n^2 - k^2, n*k - k^2) begins:
1;
1, 1;
1, 3, 1;
1, 28, 10, 1;
1, 455, 495, 35, 1;
1, 10626, 54264, 8008, 126, 1;
1, 324632, 10518300, 4686825, 125970, 462, 1;
1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1;
1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1; ...

Links

Crossrefs

Cf. A245242 (row sums), A245245 (central terms), A209330, A228832.

Programs

  • Mathematica
    Table[Binomial[n^2-k^2,n k-k^2],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 06 2019 *)
  • PARI
    {T(n,k) = binomial(n^2 - k^2, n*k - k^2)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
  • PARI
    {T(n,k) = binomial(n^2,n*k) * binomial(n*k,k^2) / binomial(n^2,k^2)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

T(n,k) = C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).
T(n,k) = (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).
T(n,k) = ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).
Showing 1-2 of 2 results.

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