A364514 - OEIS

%I #17 Oct 05 2023 15:20:27

%S 1,-2,0,16,0,-252,0,4800,0,-100100,0,2201472,0,-50139936,0,1170614016,

%T 0,-27839740500,0,671626956000,0,-16388657193480,0,403645030064640,0,

%U -10018806017062752,0,250305475771456000,0,-6288594802355952000,0,158759294846918261760

%N a(0) = 1 and a(n) = [x^n] (1 - x)^(2*n) * Legendre_P(n-1, (1 + x)/(1 - x)) for n >= 1.

%C Row 1 of A364513.

%H Paolo Xausa, <a href="/A364514/b364514.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k = 0..n} (-1)^k * binomial(n-1, n-k)^2 * binomial(n+1, k).

%F a(2*n) = 0 for n >= 1; a(2*n+1) = (-1)^(n+1) * 2/(2*n + 1) * (3*n + 1)!/n!^3.

%F a(2*n+1) ~ (-1)^(n+1) * 3^(3*n) * 3*sqrt(3)/(2*Pi*n).

%F P-recursive: a(0) = 1, a(1) = -2 and for n >= 2, a(n) = -(3*n - 1)*(3*n - 5)*(3*n - 6)/(n*(n - 1)^2) * a(n-2).

%F Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

%F a(n) = Sum_{k = 0..n} (-1)^k * binomial(n+k-1, k) * binomial(n-1, k) * binomial(2*n-k, n-k). - _Peter Bala_, Aug 13 2023

%p a := proc(n) option remember; if n = 0 then 1 elif n = 1 then -2 else -(3*n - 1)*(3*n - 5)*(3*n - 6)/(n*(n - 1)^2) * a(n-2) end if; end:

%p seq(a(n), n = 0..15);

%t A364514[n_]:=A364514[n]=Which[n==0,1,n==1,-2,True,-(3n-1)(3n-5)(3n-6)/(n(n-1)^2)A364514[n-2]];Array[A364514,40,0] (* _Paolo Xausa_, Oct 05 2023 *)

%Y Cf. A364513.

%K sign,easy

%O 0,2

%A _Peter Bala_, Aug 02 2023