A371364 Expansion of (1/x) * Series_Reversion( x * (1-4*x)^2 / (1-3*x) ).

1, 5, 49, 597, 8129, 118469, 1807665, 28512213, 461141761, 7606159365, 127454706609, 2163636679125, 37130370808257, 643099703566277, 11227141735655345, 197356077159062613, 3490230884900117505, 62054912214781757957, 1108568475427756051761

Offset: 0

Author

Seiichi Manyama, Mar 19 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A059231, A371365.

Programs

Maple

A371364 := proc(n)
    add(3^(n-k)*binomial(2*n+k+1,k)*binomial(2*n,n-k),k=0..n) ;
    %/(n+1) ;
end proc:
seq(A371364(n),n=0..60) ; # R. J. Mathar, Apr 22 2024

PARI

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

PARI

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

Formula

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

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