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.
%I A061401 #21 Oct 23 2024 00:54:16
%S A061401 1,6,9,56,-300,3942,-48412,639264,-8785098,124733000,-1817441901,
%T A061401 27047510136,-409669978497,6297861697848,-98055605511675,
%U A061401 1543575781461888,-24533312413314948,393242952308487588,-6350814254230701986
%N A061401 From solution to a Picard-Fuchs equation.
%H A061401 M. Aganagic, A. Klemm and C. Vafa, <a href="http://arXiv.org/abs/hep-th/0105045">Disk Instantons, Mirror Symmetry and the Duality Web</a>, Equation 6.15, p. 44.
%H A061401 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a061/A061401.java">Java program</a> (github)
%F A061401 From _Peter Bala_, Oct 20 2024: (Start)
%F A061401 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.
%F A061401 [x*n] (x/A(x))^n = (-1)^n * (3*n)!/n!^3 = (-1)^n * A006480(n).
%F A061401 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.
%F A061401 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)
%t A061401 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 *)
%Y A061401 Cf. A006480, A229451, A229452.
%K A061401 sign
%O A061401 1,2
%A A061401 _N. J. A. Sloane_, Jun 10 2001
%E A061401 More terms from _Jean-François Alcover_, Feb 18 2019