cp's OEIS Frontend

A386548 a(n) = [x^n] ((1 - x)/(1 - x + x^2))^n.

%I A386548 #15 Aug 03 2025 04:46:05
%S A386548 1,0,-2,-3,6,25,1,-147,-218,591,2223,-484,-14871,-18759,68353,222697,
%T A386548 -116058,-1629671,-1656989,8275203,23266031,-20154144,-184550412,
%U A386548 -141418628,1019061001,2468408775,-3122976521,-21213927840,-10837119735,126256071125,262294667301,-456407675223
%N A386548 a(n) = [x^n] ((1 - x)/(1 - x + x^2))^n.
%C A386548 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
%C A386548 Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 5 and all positive integers n and k.
%F A386548 a(n) = Sum_{k = 0..floor(n/2)} binomial(-n, k)*binomial(n-k-1, n-2*k) = Sum_{k = 0
%F A386548 ..floor(n/2)} (-1)^k*binomial(n+k-1, k)*binomial(n-k-1, n-2*k). Cf. A246437.
%F A386548 a(n) = -n*hypergeom([n+1, 1 - (1/2)*n, 3/2 - (1/2)*n], [2, 2 - n], 4) for n >= 3.
%F A386548 P-recursive: 3*n*(n - 1)*(19*n^2 - 79*n + 78)*a(n) = 2*(n - 1)*(2*n - 3)*(19*n^2 - 60*n + 36)*a(n-1) - 2*(190*n^4 - 1170*n^3 + 2519*n^2 - 2229*n + 666)*a(n-2) - 2*(n - 3)*(2*n - 3)*(19*n^2 - 41*n + 18)*a(n-3) with a(0) = 1, a(1) = 0 and a(2) = -2.
%F A386548 exp( Sum_{n >= 1} a(n)*(-x)^n/n ) = 1 - x^2 + x^3 + 2*x^4 - 6*x^5 - x^6 +  ... is the g.f. of A364374.
%p A386548 a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 0 elif n = 2 then -2 else
%p A386548 ( 2*(n-1)*(2*n-3)*(19*n^2-60*n+36)*a(n-1) - 2*(190*n^4-1170*n^3+2519*n^2-2229*n+666)*a(n-2) - 2*(n-3)*(2*n-3)*(19*n^2-41*n+18)*a(n-3) )/(3*n*(n-1)*(19*n^2-79*n+78)) fi; end:
%p A386548 seq(a(n), n = 0..30);
%t A386548 a[n_]:=SeriesCoefficient[((1 - x)/(1 - x + x^2))^n,{x,0,n}]; Array[a,32,0] (* _Stefano Spezia_, Jul 29 2025 *)
%o A386548 (PARI) a(n) = my(x='x+O('x^(n+1))); polcoef(((1 - x)/(1 - x + x^2))^n, n); \\ _Michel Marcus_, Aug 03 2025
%Y A386548 Cf. A104507, A246437, A364374, A370616.
%K A386548 sign,easy
%O A386548 0,3
%A A386548 _Peter Bala_, Jul 25 2025