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 A364299 #8 Jul 20 2023 10:08:59 %S A364299 1,1,19,721,49251,5370751,859748023,190320431953,55743765411043, %T A364299 20884452115700251,9745388924112505269,5543574376457462884111, %U A364299 3776677001062829977964007,3036161801705682492174749691,2844274879825369072829081331519 %N A364299 a(n) = [x^n] 1/(1 + x) * Legendre_P(n, (1 - x)/(1 + x))^(-1) for n >= 0. %C A364299 Row 1 of A364298. %C A364299 Compare with the Apéry numbers A005258, which are related to the Legendre polynomials by A005258(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x)). %C A364299 A005258 satisfies the supercongruences %C A364299 1) u (n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) %C A364299 and the shifted supercongruences %C A364299 2) u (n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r)) %C A364299 for all primes p >= 5 and positive integers n and r. %C A364299 We conjecture that the present sequence also satisfies the supercongruences 1) and 2). %F A364299 Conjectures: %F A364299 1) 13*a(p) - 7*a(p-1) == 6 (mod p^5) for all primes p >= 3 (checked up to p = 101). %F A364299 2) for r >= 2, 13*a(p^r) - 7*a(p^r - 1) == 13*a(p^(r-1)) - 7*a(p^(r-1) - 1) (mod p^(3*r+3)) for all primes p >= 5. %F A364299 3) a(p)^13 == a(p-1)^7 (mod p^5) for all primes p >= 3 (checked up to p = 101). %F A364299 4) for r >= 2, a(p^r)^13 * a(p^(r-1) - 1)^7 == a(p^(r-1))^13 * a(p^r - 1)^7 (mod p^(3*r+3)) for all primes p >= 5. %p A364299 a(n) := coeff(series( 1/(1 + x) * LegendreP(n, (1 - x)/(1 + x))^(-1), x, 21), x, n): %p A364299 seq(a(n), n = 0..20); %Y A364299 Cf. A005258, A364298, A364300, A364301, A364302. %K A364299 nonn,easy %O A364299 0,3 %A A364299 _Peter Bala_, Jul 18 2023