A390859 - OEIS

%I #14 Nov 26 2025 12:12:41

%S 1,3,6,12,26,57,123,264,568,1224,2637,5679,12230,26340,56730,122181,

%T 263143,566736,1220592,2628816,5661737,12193803,26262054,56561148,

%U 121816954,262359777,565049859,1216959960,2620992680,5644887960,12157515861,26183901783,56392828966

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

%H Vincenzo Librandi, <a href="/A390859/b390859.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,-1).

%F G.f.: 1/((1-x)^4 * (1+g-g^2)), where g = x/(1-x)^2.

%F G.f.: 1 / (1 - 3*x + 3*x^2 - 3*x^3 + x^4).

%F a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - a(n-4).

%t Table[Sum[(-1)^k*Binomial[n+k+3,n-k]*Fibonacci[k+1],{k,0,n}],{n,0,40}] (* _Vincenzo Librandi_, Nov 26 2025 *)

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

%o (Magma) [&+[(-1)^k*Binomial(n+k+3, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // _Vincenzo Librandi_, Nov 26 2025

%Y Partial sums of A390853.

%Y Cf. A000045, A099444.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Nov 21 2025