A391035 - OEIS

%I #13 Dec 02 2025 03:36:50

%S 1,1,2,5,15,47,152,504,1706,5872,20490,72319,257723,926071,3351552,

%T 12205864,44698318,164492726,608009172,2256266234,8402790998,

%U 31395570766,117652696392,442095949780,1665396815980,6288134065462,23793282441962,90208829415539,342646535562835

%N Expansion of g/(1 - x^4*g^4), where g = 1+x*g^2 is the g.f. of A000108.

%H Vincenzo Librandi, <a href="/A391035/b391035.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/4)} (4*k+1) * binomial(2*n-4*k+1,n-4*k)/(2*n-4*k+1).

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (4*k+1) * binomial(2*n-4*k,n-4*k).

%t Table[Sum[(4*k+1)*Binomial[2*n-4*k+1,n-4*k]/(2*n-4*k+1),{k,0,Floor[n/4]}],{n,0,30}] (* _Vincenzo Librandi_, Dec 01 2025 *)

%o (PARI) a(n) = sum(k=0, n\4, (4*k+1)*binomial(2*n-4*k+1, n-4*k)/(2*n-4*k+1));

%o (Magma) [&+[(4*k+1)*Binomial(2*n-4*k+1, n-4*k)/(2*n-4*k+1): k in [0..Floor(n/4)]] : n in [0..40] ]; // _Vincenzo Librandi_, Dec 01 2025

%Y Cf. A390479, A391030, A391032, A391034.

%Y Cf. A000108.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 26 2025