A171106 Number of irreducible rational curves maximally tangent to a line.

1, 1, 7, 138, 5477, 367640, 37541883, 5432772352, 1059075055273, 267757626501504, 85244466165571535, 33379687015338236672, 15770655073870516443597, 8847780392111931116474368, 5815426547948880787678282627, 4426738320076692932937846865920, 3863757059392099045703984747480273

Offset

1,3

Comments

Possibly related to left column of table 2 in Bousseau-Wu. - F. Chapoton, Feb 20 2024

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..222

Pierrick Bousseau and Longting Wu, All-Genus WDVV Recursion, Quivers, and BPS Invariants, arXiv:2303.00503, 2023.

Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828 [math.AG], 2009-2010.

Formula

Sergey Fomin and Grigory Mikhalkin give a recurrence.

From Gheorghe Coserea, Jan 04 2018: (Start)

(n-1)*(2*n-1)*n^2 * a(n) = 2*s1(n) - s2(n) - s3(n) + 5*s4(n) - 3*s5(n), where s1(n) = Sum_{i+j+k=n} i^3*j^3*k^3 * (2*n)!/((2*i)!*(2*j)!*(2*k)!) * a(i)*a(j)*a(k), s2(n) = Sum_{i+j+k=n} i^3*j^3*k^2 * (2*n)!/((2*i)!*(2*j)!*(2*k)!) * a(i)*a(j)*a(k), s3(n) = Sum_{i+j=n} i^3*(i-1)*j^2 * (2*n)!/((2*i)!*(2*j)!) * a(i)*a(j), s4(n) = Sum_{i+j=n} i^3*j^3 * (2*n)!/((2*i)!*(2*j)!) * a(i)*a(j), s5(n) = Sum_{i+j=n} i^3*j^2 * (2*n)!/((2*i)!*(2*j)!) * a(i)*a(j) , n>=3.

x*(4*y' - exp(y) - x*exp(y)*y') = 2*y, where y(x) = Sum_{n>=1} n^2*a(n)/(2*n)! * x^n. (eqn. (4.8))

(End)

a(n) ~ c * d^n * n^(2*n-3), where d = 0.520598617698520888136853407904080840281... and c = 1.8295420351460716385409955906837076169... - Vaclav Kotesovec, Oct 24 2018

Mathematica

terms = 17; Clear[a]; a[1] = a[2] = 1;
y[x_] = Sum[d^2 a[d] x^d/(2d)!, {d, 1, terms+1}];
f = x (4y'[x] - E^y[x] - x E^y[x] y'[x]) - 2y[x] + O[x]^(terms+1);
Solve[0 == Thread[CoefficientList[f, x] ]][[1]] /. Rule -> Set;
Array[a, terms] (* Jean-François Alcover, Oct 24 2018 *)

Prog

(PARI)
runs(v) = {
  my(a=List(), j=1, k=1);
  for (i=2, #v, if (v[i] == v[j], k++, listput(a, k); k=1; j=i));
  listput(a, k); Vec(a);
};
M(v) = vecsum(v)! \ prod(k=1, #v, v[k]!);
A171106_seq(N) = {
  my(a = vector(N), s, p, q); a[1]=1; a[2]=1;
  for (n=2, N-1, a[n+1] = sum(k=1, n, s=0;
    forpart(p=n, q = prod(i=1, k, p[i]^2*a[p[i]]/(2*p[i])!);
      s += q * M(runs(p)), , [k, k]); (2*n)!/k! * s)); a;
};
A171106_seq(17)
\\ test: N=33; a=A171106_seq(N); y='x*Ser(vector(N, n, n^2*a[n]/(2*n)!)); x*(4*y' - exp(y) - x*exp(y)*y') == 2*y
(PARI)
A171106_seq(N) = {
  my(a=vector(N), s1, s2, s3, s4=vector(N), s5);
  a[1]=1; a[2]=1; s4[1]=0; s4[2]=6;
  for (n=3, N,
    s4[n] = sum(k=1, n-1, k^3*(n-k)^3 * binomial(2*n, 2*k) * a[k]*a[n-k]);
    s1 = sum(k=1, n-1, k^3*binomial(2*n, 2*k) * s4[n-k]*a[k]);
    s2 = sum(k=1, n-1, k^2*binomial(2*n, 2*k) * s4[n-k]*a[k]);
    s3 = sum(k=1, n-1, k^3*(k-1)*(n-k)^2 * binomial(2*n, 2*k) * a[k]*a[n-k]);
    s5 = sum(k=1, n-1, k^3*(n-k)^2 * binomial(2*n, 2*k) * a[k]*a[n-k]);
    a[n] = (2*s1 - s2 - s3 + 5*s4[n] - 3*s5) / ((n-1)*(2*n-1)*n^2)); a;
};
A171106_seq(17) \\ Gheorghe Coserea, Jan 04 2018

Crossrefs

Cf. A171107.

Sequence in context: A221298 A009701 A120052 * A217504 A142295 A056254

Adjacent sequences: A171103 A171104 A171105 * A171107 A171108 A171109

Keyword

nonn

Author

N. J. A. Sloane, Sep 27 2010

Extensions

Terms a(15) and beyond from Gheorghe Coserea, Jan 02 2018

Status

approved