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A350406 a(n) = [x^n] 1/(1 + x + x^2 + x^3)^n.

%I A350406 #29 Mar 19 2023 05:36:02
%S A350406 1,-1,1,-1,5,-26,91,-246,597,-1540,4576,-14521,44915,-132328,380290,
%T A350406 -1102076,3268437,-9838428,29616364,-88538500,263489380,-785026110,
%U A350406 2348923875,-7053379710,21204016275,-63716916276,191394838116,-575200476046,1730575897202
%N A350406 a(n) = [x^n] 1/(1 + x + x^2 + x^3)^n.
%H A350406 Seiichi Manyama, <a href="/A350406/b350406.txt">Table of n, a(n) for n = 0..1000</a>
%F A350406 a(n) = (-1)^n * Sum_{k=0..n} binomial(n-1+k,k) * binomial(n,4*k).
%F A350406 From _Peter Bala_, Mar 17 2023: (Start)
%F A350406 a(n) = Sum_{k = 0..floor(n/2)} (-1)^(n-k)*binomial(n+k-1,k)*binomial(2*n-2*k-1,n-2*k).
%F A350406 16*n*(n+1)*(n-1)*(5*n-4)*(5*n-8)*(5*n-9)*a(n) = - ( 4*n*(n-1)*(5*n-9)* (550*n^3-1045*n^2+164*n+223)*a(n-1) + (n-1)*(5*n+1)*(5075*n^4-22330*n^3 +33771*n^2-20232*n+4032)*a(n-2) + 8*(2*n-3)*(4*n-5)*(4*n-7)*(5*n+1)*(5*n-3)*(5*n-4)*a(n-3) ).
%F A350406 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. (End)
%F A350406 a(n) = (-1)^n*hypergeom([1/4 - n/4, 1/2 - n/4, 3/4 - n/4, -n/4, n], [1/4, 1/2, 3/4, 1], 1). - _Peter Luschny_, Mar 19 2023
%p A350406 a := proc (n) option remember; if n = 0 then 1 elif n = 1 then -1 elif n = 2 then 1 else -(4*n*(n-1)*(5*n-9)*(550*n^3-1045*n^2+164*n+223)*a(n-1) + (n-1)*(5*n+1)*(5075*n^4-22330*n^3+33771*n^2-20232*n+4032)*a(n-2) + (16*n-24)*(4*n-5)*(4*n-7)*(5*n+1)*(5*n-3)*(5*n-4)*a(n-3))/(16*n*(n+1)*(n-1)*(5*n-4)*(5*n-8)*(5*n-9)) end if; end:
%p A350406 seq(a(n), n = 0..30); # _Peter Bala_, Mar 18 2023
%p A350406 a := n -> (-1)^n*hypergeom([1/4 - n/4, 1/2 - n/4, 3/4 - n/4, -n/4, n], [1/4, 1/2, 3/4, 1], 1): seq(simplify(a(n)), n = 0..28); # _Peter Luschny_, Mar 19 2023
%t A350406 a[n_] := Coefficient[Series[1/(1 + x + x^2 + x^3)^n, {x, 0, n}], x, n]; Array[a, 30, 0] (* _Amiram Eldar_, Dec 29 2021 *)
%o A350406 (PARI) a(n) = (-1)^n*sum(k=0, n, binomial(n-1+k, k)*binomial(n, 4*k));
%Y A350406 Cf. A005725, A348410, A350383, A350407.
%K A350406 sign,easy
%O A350406 0,5
%A A350406 _Seiichi Manyama_, Dec 29 2021