A350383 - OEIS

%I #56 Mar 12 2025 08:23:04

%S 1,-1,1,2,-15,49,-98,48,561,-2860,8151,-12948,-9282,149226,-594320,

%T 1428952,-1448655,-5538975,37450900,-122995950,239589735,-37528755,

%U -1886983020,8939152560,-24579514050,35197176924,51580335366,-541312482256,2033695030128,-4624358661240

%N a(n) = [x^n] 1/(1 + x + x^2)^n.

%H Seiichi Manyama, <a href="/A350383/b350383.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1+k,k) * binomial(n,3*k).

%F Recurrence: 3*(n-1)*n*(4*n - 7)*a(n) = -2*(n-1)*(28*n^2 - 63*n + 27)*a(n-1) - 3*(3*n - 5)*(3*n - 4)*(4*n - 3)*a(n-2). - _Vaclav Kotesovec_, Mar 18 2023

%F From _Peter Bala_, Apr 15 2023: (Start)

%F a(n) = (-1)^n*hypergeom([-n/3, 1/3 - n/3, 2/3 - n/3, n], [1/3, 2/3, 1], 1).

%F Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. Cf. A228960.

%F More generally, let k be a positive integer, m an integer and let f(x) = g(x)/h(x), where g(x) and h(x) are both finite products of cyclotomic polynomials. Then we conjecture that the same supercongruences hold, except for a finite number of primes p depending on f(x), for the sequence {a_(k,m,f)(n): n >= 0} defined by a_(k,m,f)(n) = [x^(k*n)] f(x)^(m*n). (End)

%F From _Peter Bala_, Mar 11 2025: (Start)

%F G.f.: A(x) = 1 + x*d/dx(log(G(x)/x)), where G(x) = x - x^2 + x^3 - 4*x^5 + 14*x^6 - 30*x^7 + ... is the g.f. of A103779.

%F The following formulas hold for n >= 1:

%F a(n) = [x^n] T(2*n, (1 - x)/2), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.

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

%F a(n) = (1/2)*(-1)^n*binomial(2*n, n)*hypergeom([-n/2, (-n+1)/2], [-2*n+1], 4). Cf. A213684. (End)

%p a := n -> (-1)^n*hypergeom([-n/3, 1/3 - n/3, 2/3 - n/3, n], [1/3, 2/3, 1], 1): seq(simplify(a(n)), n = 0..30); # _Peter Bala_, Apr 17 2023

%t a[n_] := Coefficient[Series[1/(1 + x + x^2)^n, {x, 0, n}], x, n]; Array[a, 30, 0] (* _Amiram Eldar_, Dec 29 2021 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n-1+k, k)*binomial(n, 3*k));

%Y Cf. A002426, A027907, A103779, A110556, A165817, A213684, A228960, A350406, A350407, A362087.

%K sign,easy,changed

%O 0,4

%A _Seiichi Manyama_, Dec 29 2021