A390713 - OEIS

%I #17 Nov 18 2025 05:43:14

%S 1,1,3,14,81,528,3706,27343,209091,1642505,13175592,107475228,

%T 888788286,7434421879,62790108909,534724808122,4586532562769,

%U 39588016078536,343595160001123,2996862925155061,26254154919371088,230913398343788951,2038243761068156956

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

%H Vincenzo Librandi, <a href="/A390713/b390713.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(1-x*g^2) where g = 1+x*g^4 is the g.f. of A002293.

%F a(n) = binomial(4*n-2, n-1)*hypergeom([2, 1-3*n, 1-n], [(3-4*n)/2, 2*(1-n)], 1/2^2)/(2*n - 1) for n > 0. - _Stefano Spezia_, Nov 16 2025

%t a[n_]:=Binomial[4*n-2,n-1]*HypergeometricPFQ[{2,1-3n,1-n},{(3-4n)/2, 2(1-n)}, 1/4]/(2n-1); Join[{1},Array[a,22]] (* _Stefano Spezia_, Nov 16 2025 *)

%t Join[{1},Table[Sum[k*Binomial[4*n-2*k,n-k]/(2*n-k),{k,0,n}],{n,1,25}]] (* _Vincenzo Librandi_, Nov 18 2025 *)

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

%o (Magma) [1] cat [&+[k*Binomial(4*n-2*k, n-k)/(2*n-k): k in [0..n]] : n in [1..30] ]; // _Vincenzo Librandi_, Nov 18 2025

%Y Cf. A002293, A130458, A390714.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 15 2025