A386774 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^3 / (1+2*x)^4 ).

1, 17, 439, 13513, 458196, 16518407, 621247194, 24099952473, 957294067516, 38741943503972, 1591753835634799, 66219447135668383, 2783826043226606236, 118078452737821009962, 5047034289902290964004, 217173909723115943823993, 9400092428228971114597356

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..596

Index entries for reversions of series

Formula

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

a(n) = (1/(n+1)) * [x^n] ( (1+2*x)^4 / (1-3*x)^3 )^(n+1).

D-finite with recurrence: (6973030400*n^3 + 31378636800*n^2 + 46632140800*n + 22880256000)*a(n) + (9779650560*n^3 + 36426792960*n^2 + 33279836160*n - 2647142400)*a(n + 1) + (-3299885568*n^3 - 27310556160*n^2 - 74594794752*n - 67228341120)*a(n + 2) + (90384336*n^3 + 965569248*n^2 + 3418228512*n + 4007721240)*a(3 + n) + (-531441*n^3 - 7440174*n^2 - 34661763*n - 53734590)*a(n + 4) = 0. - Robert Israel, Mar 12 2026

Maple

f:= gfun:-rectoproc({(6973030400*n^3 + 31378636800*n^2 + 46632140800*n + 22880256000)*a(n) + (9779650560*n^3 + 36426792960*n^2 + 33279836160*n - 2647142400)*a(n + 1) + (-3299885568*n^3 - 27310556160*n^2 - 74594794752*n - 67228341120)*a(n + 2) + (90384336*n^3 + 965569248*n^2 + 3418228512*n + 4007721240)*a(3 + n) + (-531441*n^3 - 7440174*n^2 - 34661763*n - 53734590)*a(n + 4), a(0) = 1, a(1) = 17, a(2) = 439, a(3) = 13513}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Mar 12 2026

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-3*x)^3/(1+2*x)^4)/x)
(PARI) a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(4*(n+1), k)*binomial(4*n-k+2, n-k))/(n+1);

Crossrefs

Cf. A386723, A386768, A386771.

Sequence in context: A158063 A370543 A301642 * A035022 A375891 A012117

Adjacent sequences: A386771 A386772 A386773 * A386775 A386776 A386777

Keyword

nonn

Author

Seiichi Manyama, Aug 02 2025

Status

approved