A061401 From solution to a Picard-Fuchs equation.

1, 6, 9, 56, -300, 3942, -48412, 639264, -8785098, 124733000, -1817441901, 27047510136, -409669978497, 6297861697848, -98055605511675, 1543575781461888, -24533312413314948, 393242952308487588, -6350814254230701986

Offset

1,2

Links

Table of n, a(n) for n=1..19.

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, Equation 6.15, p. 44.

Sean A. Irvine, Java program (github)

Formula

From Peter Bala, Oct 20 2024: (Start)

The g.f. A(x) = x + 6*x^2 + 9*x^3 + 56*x^4 - 300*x^5 + ... = x*series_reversion(B(x)), where B(x) = exp( Sum_{n >= 1} (-1)^n*(3*n)!/n!^3*x^n/n ). See A229451.

[x*n] (x/A(x))^n = (-1)^n * (3*n)!/n!^3 = (-1)^n * A006480(n).

The power series F(x) := (A(x)/x)^(1/6) = 1 + x - x^2 + 11*x^3 - 100*x^4 + 1101*x^5 - 13273*x^6 + 170860*x^7 - 2306884*x^8 + 32300950*x^9 - 465426461*x^10 + ... appears to have integer coefficients.

Conjecture: Let r be an integer and s a positive integer. The sequence defined by u(n) = [x^(s*n)] F(x)^(r*n) satisfies the supercongruence u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r. (End)

Mathematica

InverseSeries[z/Exp[6 z HypergeometricPFQ[{1, 1, 4/3, 5/3}, {2, 2, 2}, -27 z]] + O[z]^20, q] // CoefficientList[#, q]& // Rest (* Jean-François Alcover, Feb 18 2019 *)

Crossrefs

Cf. A006480, A229451, A229452.

Sequence in context: A098662 A056425 A056416 * A359800 A264380 A284636

Adjacent sequences: A061398 A061399 A061400 * A061402 A061403 A061404

Keyword

sign

Author

N. J. A. Sloane, Jun 10 2001

Extensions

More terms from Jean-François Alcover, Feb 18 2019

Status

approved