A171106 - OEIS

%I #47 Feb 20 2024 14:40:42

%S 1,1,7,138,5477,367640,37541883,5432772352,1059075055273,

%T 267757626501504,85244466165571535,33379687015338236672,

%U 15770655073870516443597,8847780392111931116474368,5815426547948880787678282627,4426738320076692932937846865920,3863757059392099045703984747480273

%N Number of irreducible rational curves maximally tangent to a line.

%C Possibly related to left column of table 2 in Bousseau-Wu. - _F. Chapoton_, Feb 20 2024

%H Gheorghe Coserea, <a href="/A171106/b171106.txt">Table of n, a(n) for n = 1..222</a>

%H Pierrick Bousseau and Longting Wu, <a href="https://arxiv.org/abs/2303.00503">All-Genus WDVV Recursion, Quivers, and BPS Invariants</a>, arXiv:2303.00503, 2023.

%H Sergey Fomin and Grigory Mikhalkin, <a href="https://arxiv.org/abs/0906.3828">Labeled floor diagrams for plane curves</a>, arXiv:0906.3828 [math.AG], 2009-2010.

%F Sergey Fomin and Grigory Mikhalkin give a recurrence.

%F From _Gheorghe Coserea_, Jan 04 2018: (Start)

%F (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.

%F 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))

%F (End)

%F a(n) ~ c * d^n * n^(2*n-3), where d = 0.520598617698520888136853407904080840281... and c = 1.8295420351460716385409955906837076169... - _Vaclav Kotesovec_, Oct 24 2018

%t terms = 17; Clear[a]; a[1] = a[2] = 1;

%t y[x_] = Sum[d^2 a[d] x^d/(2d)!, {d, 1, terms+1}];

%t f = x (4y'[x] - E^y[x] - x E^y[x] y'[x]) - 2y[x] + O[x]^(terms+1);

%t Solve[0 == Thread[CoefficientList[f, x] ]][[1]] /. Rule -> Set;

%t Array[a, terms] (* _Jean-François Alcover_, Oct 24 2018 *)

%o (PARI)

%o runs(v) = {

%o   my(a=List(),j=1,k=1);

%o   for (i=2, #v, if (v[i] == v[j], k++, listput(a,k); k=1; j=i));

%o   listput(a,k); Vec(a);

%o };

%o M(v) = vecsum(v)! \ prod(k=1, #v, v[k]!);

%o A171106_seq(N) = {

%o   my(a = vector(N),s,p,q); a[1]=1; a[2]=1;

%o   for (n=2, N-1, a[n+1] = sum(k=1, n, s=0;

%o     forpart(p=n, q = prod(i=1, k, p[i]^2*a[p[i]]/(2*p[i])!);

%o       s += q * M(runs(p)), , [k,k]); (2*n)!/k! * s)); a;

%o };

%o A171106_seq(17)

%o \\ 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

%o (PARI)

%o A171106_seq(N) = {

%o   my(a=vector(N), s1, s2, s3, s4=vector(N), s5);

%o   a[1]=1; a[2]=1; s4[1]=0; s4[2]=6;

%o   for (n=3, N,

%o     s4[n] = sum(k=1, n-1, k^3*(n-k)^3 * binomial(2*n, 2*k) * a[k]*a[n-k]);

%o     s1 = sum(k=1, n-1, k^3*binomial(2*n, 2*k) * s4[n-k]*a[k]);

%o     s2 = sum(k=1, n-1, k^2*binomial(2*n, 2*k) * s4[n-k]*a[k]);

%o     s3 = sum(k=1, n-1, k^3*(k-1)*(n-k)^2 * binomial(2*n, 2*k) * a[k]*a[n-k]);

%o     s5 = sum(k=1, n-1, k^3*(n-k)^2 * binomial(2*n, 2*k) * a[k]*a[n-k]);

%o     a[n] = (2*s1 - s2 - s3 + 5*s4[n] - 3*s5) / ((n-1)*(2*n-1)*n^2)); a;

%o };

%o A171106_seq(17) \\ _Gheorghe Coserea_, Jan 04 2018

%Y Cf. A171107.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Sep 27 2010

%E Terms a(15) and beyond from _Gheorghe Coserea_, Jan 02 2018