A390147 - OEIS

%I #38 May 20 2026 19:58:42

%S 1,2,10,56,363,2540,18719,143084,1123888,9015938,73553356,608356076,

%T 5089508859,42992095102,366179399391,3141310120084,27117461481521,

%U 235389702516728,2053329861556631,17990275294687148,158246298370021478,1396954300222929984

%N Expansion of g^2/(1 - x^2*g^2), where g = 1+x*g^4 is the g.f. of A002293.

%H Vincenzo Librandi, <a href="/A390147/b390147.txt">Table of n, a(n) for n = 0..1000</a>

%H Robert Israel, <a href="/A390147/a390147.txt">Linear recurrence of order 16</a>

%F a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(4*n-6*k+2,n-2*k)/(2*n-3*k+1).

%F D-finite with recurrence of order 16 (see link). - _Robert Israel_, May 20 2026

%t Table[Sum[ (k+1)*Binomial[4* n-6*k+2,n-2*k]/(2*n-3*k+1),{k,0,Floor[n/2]}],{n,0,26}] (* _Vincenzo Librandi_, Nov 29 2025 *)

%o (PARI) a(n) = sum(k=0, n\2, (k+1)*binomial(4*n-6*k+2, n-2*k)/(2*n-3*k+1));

%o (Magma) [&+[(k+1)*Binomial(4*n-6*k+2, n-2*k)/(2*n-3*k+1): k in [0..Floor(n/2)]] : n in [0..40] ]; // _Vincenzo Librandi_, Nov 29 2025

%Y Cf. A389112, A391080, A391082, A391084.

%Y Cf. A002293.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 27 2025