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A363984 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n,k)*binomial(n+k,k)*A363983(k).

%I A363984 #14 Jul 10 2023 02:09:48
%S A363984 1,3,73,2163,75001,2835003,113329945,4711519347,201638246905,
%T A363984 8824346685003,393088036809073,17764622316152715,812477640612743977,
%U A363984 37535247213943518315,1749047441756088054073,82108960863923963522163,3879675478363506548275705
%N A363984 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n,k)*binomial(n+k,k)*A363983(k).
%C A363984 The Legendre transform of a sequence {b(n)} is the sequence {c(n)} defined by c(n) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k,k)*b(k).
%C A363984 It is known that the sequence of Apéry numbers A005259 is the Legendre transform of the sequence of Franel numbers A000172. Note that A000172(n) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(2*k,n) (the first Strehl identity). Here we consider the Legendre transform of (a signed version of) A363983, which by analogy with Strehl's identity for A000172 is given by A363981(n) = (-1)^n * Sum_{k = 0..n} binomial(-n,k)* binomial(n,k)*binomial(2*k,n).
%C A363984 The Apéry numbers A005259 and also A005258 satisfy the pair of supercongruences (see, for example, Straub, Introduction)
%C A363984 (1) u(n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r)) and
%C A363984 (2) u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)),
%C A363984 with both congruences valid for all primes p >= 5 and all positive integers n and r.
%C A363984 Straub, Example 3.4, observes that, among the known Apéry-like numbers, the Apéry numbers A005258 and A005259 are the only ones to satisfy shifted supercongruences of the form (1) in addition to the supercongruences of the form (2).
%C A363984 We conjecture that the present sequence satisfies the same pair of congruences.
%H A363984 Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.
%H A363984 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendreTransform.html">Legendre Transform</a>
%H A363984 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StrehlIdentities.html">Strehl identities</a>
%e A363984 Examples of supercongruences:
%e A363984 p = 11:
%e A363984 a(11) - a(1) = 17764622316152715 - 3 = (2^3)*(3^2)*7*(11^3)*13*2037061001 == 0 (mod 11^3).
%e A363984 a(11 - 1) - a(0) = 393088036809073 - 1 = (2^4)*3*(11^3)*29*67*1381*2293 == 0 (mod 11^3).
%e A363984 p = 5:
%e A363984 a(5^2) - a(5) = 5545311482504558271924122566108960335003 - 2835003 = (2^4)*3*(5^7)*11*31*91546780597609*23684663949545369 == 0 (mod 5^7).
%e A363984 a(5^2 - 1) - a(5 - 1) = 113353062539459038723143413569578825001 - 75001 = (2^4)*3*(5^7)*(11^2)*29*53*162533449533306503812325773 == 0 (mod 5^7).
%p A363984 A363983 := proc(n) option remember: add((-1)^(n+k)*binomial(n,k)*binomial(n+k-1,k)*binomial(2*k,n), k = 0..n) end:
%p A363984 seq(add((-1)^(n+k)*binomial(n,k)*binomial(n+k,k)*A363983(k), k = 0..n), n = 0..20);
%Y A363984 Cf. A000172, A005259, A363983.
%K A363984 nonn,easy
%O A363984 0,2
%A A363984 _Peter Bala_, Jul 01 2023