A371364 - OEIS

%I #15 Apr 22 2024 12:48:45

%S 1,5,49,597,8129,118469,1807665,28512213,461141761,7606159365,

%T 127454706609,2163636679125,37130370808257,643099703566277,

%U 11227141735655345,197356077159062613,3490230884900117505,62054912214781757957,1108568475427756051761

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

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

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

%F D-finite with recurrence (n+1)*(2*n+1)*a(n) +3*(-6*n^2-9*n+2)*a(n-1) -27*(7*n-9)*(2*n-3)*a(n-2) -243*(n-2)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Apr 22 2024

%p A371364 := proc(n)

%p     add(3^(n-k)*binomial(2*n+k+1,k)*binomial(2*n,n-k),k=0..n) ;

%p     %/(n+1) ;

%p end proc:

%p seq(A371364(n),n=0..60) ; # _R. J. Mathar_, Apr 22 2024

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-4*x)^2/(1-3*x))/x)

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

%Y Cf. A059231, A371365.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 19 2024