A382649 - OEIS

%I #13 May 16 2025 19:29:00

%S 1,2,15,52,213,834,3043,11576,41601,152458,544039,1950132,6895773,

%T 24403302,85542339,300101048,1044436937,3639851814,12594713911,

%U 43660404108,150357976533,518991977194,1780132570723,6122965091976,20928650616113,71779065646510,244590689773839

%N Expansion of 1/(1 - x*(1 + 4*x)^(3/2))^2.

%C a(82) is negative.

%H Vincenzo Librandi, <a href="/A382649/b382649.txt">Table of n, a(n) for n = 0..600</a>

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

%t Table[Sum[4^(n-k)* (k+1)* Binomial[3*k/2, n-k],{k,0,n}],{n,0,28}] (* _Vincenzo Librandi_, May 13 2025 *)

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

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 28); Coefficients(R!( 1/(1 - x*(1 + 4*x)^(3/2))^2)); // _Vincenzo Librandi_, May 13 2025

%Y Cf. A382536, A382650.

%K sign,easy

%O 0,2

%A _Seiichi Manyama_, Apr 02 2025