A364973 a(n) = number of degree n rational curves in the complex projective plane which satisfy an order 3n-1 local tangency constraint.

1, 1, 4, 26, 217, 2110, 22744, 264057, 3242395, 41596252, 552733376, 7559811021, 105919629403, 1514674166755, 22043665219240, 325734154669786, 4877954835706120, 73914684068584441, 1131820243084746628, 17494508772311055354, 272708269111411142397, 4283702718045699720655

Offset

1,3

Comments

This is the number of degree n rational curves in the complex projective plane which pass through a specified point and have contact order 3n-1 to a generic smooth local divisor through that point.

After multiplying by 3n-1, this sequence appears to agree with A353195.

Computed in terms of Gromov-Witten invariants of rational surfaces in McDuff-Siegel 2021.

Agrees with the count of osculating curves, which are defined and computed by a recursive formula in Muratore 2021.

Computed by a recursive formula in Mikhalkin-Siegel 2023.

Computed by a closed formula as sum over trees with n leaves and combinatorially defined weights in Siegel 2023.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..108

D. McDuff and K. Siegel, Counting curves with local tangency constraints, J. Top., 14 (2021) 1176-1242.

G. Mikhalkin and K. Siegel, Ellipsoidal superpotentials and stationary descendants, arXiv:2307.13252 [math.SG] (2023).

G. Muratore, A recursive formula for osculating curves, Arkiv Mat., 59 (2021) 195-211.

K. Siegel, Computing higher symplectic capacities I, Int. Math. Res. Not., 2022 (2021) 12402-12461.

K. Siegel, A tree formula for the ellipsoidal superpotential of the complex projective plane (2023).

Prog

(Sage)
# The following code is written in Sage and is based on the recursive formula in Mikhalkin-Siegel 2023. Note that a slightly more efficient formula is given by using Partitions instead of OrderedPartitions, at the cost of an extra combinatorial factor.
def a(n):
	if n == 1:
		return 1
	out = 1/(factorial(n)**3)
	for p in [p for p in OrderedPartitions(n) if len(p) > 1]:
		k = len(p)
		summand = prod([(3*m-1)*a(m) for m in p])
		summand /= factorial(k)*factorial(3*n-k)
		out -= summand
	out *= factorial(3*n-2)
	return out
for n in range(1, 20):
	print(a(n))
(Python)
from functools import lru_cache
from fractions import Fraction
from math import prod, factorial
from sympy.utilities.iterables import partitions
@lru_cache(maxsize=None)
def A364973(n): # after Sage code
    return 1 if n==1 else int(factorial(3*n-2)*(Fraction(1, factorial(n)**3)-sum(Fraction(prod(((3*m-1)*A364973(m))**e for m, e in p.items()), factorial(3*n-s)*prod(factorial(c) for c in p.values())) for s, p in partitions(n, k=n-1, size=True)))) # Chai Wah Wu, Nov 10 2023

Crossrefs

Cf. A353195 (after multiplying by 3n-1).

Sequence in context: A224734 A143436 A135884 * A120971 A187826 A145347

Adjacent sequences: A364970 A364971 A364972 * A364974 A364975 A364976

Keyword

nonn

Author

Kyler Siegel, Aug 22 2023

Extensions

More terms from Chai Wah Wu, Nov 10 2023

Status

approved