A393407 Expansion of (1/x) * Series_Reversion( x * (3/(1 + x)^3 - 2) ).

1, 9, 144, 2865, 63783, 1520946, 37989600, 981163395, 25989236235, 702155078427, 19274174370288, 536030361923004, 15071094643290504, 427685151005699670, 12233798976412720512, 352372368517249479735, 10211166472432551608871, 297492551409510590947551

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

G.f. A(x) satisfies A(x) = 1/(3/(1 + x*A(x))^3 - 2).

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

D-finite with recurrence 648*n*(n-1)*(1561*n-2712)*(n+1)*a(n) -36*n*(n-1)*(888797*n^2 -1945293*n +680454)*a(n-1) -24*(n-1)*(201243*n^3 -2828358*n^2 +8627387*n -7326114)*a(n-2) +4*(117495*n^4 -3457926*n^3 +18898361*n^2 -36809446*n+23851740)*a(n-3) +2*(-266441*n^4 +3705239*n^3 -17988251*n^2 +35736017*n-24111780)*a(n-4) +(3*n-11)*(10339*n-14692) *(n-5)*(3*n-10)*a(n-5)=0. - R. J. Mathar, Feb 27 2026

Prog

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

Crossrefs

Cf. A103210, A231552.

Cf. A393380.

Sequence in context: A134176 A325173 A067415 * A069134 A034829 A162993

Adjacent sequences: A393404 A393405 A393406 * A393408 A393409 A393410

Keyword

nonn

Author

Seiichi Manyama, Feb 13 2026

Status

approved