A369592 Expansion of (1/x) * Series_Reversion( x / (1+x+x^4/(1+x)^3) ).

1, 1, 1, 1, 2, 3, 4, 5, 10, 16, 23, 31, 62, 102, 152, 213, 426, 712, 1084, 1556, 3112, 5255, 8116, 11843, 23686, 40288, 62866, 92842, 185684, 317548, 499376, 744277, 1488554, 2556376, 4044740, 6072124, 12144248, 20926236, 33272912, 50244660, 100489320, 173634752

Offset

0,5

Comments

Number of ordered trees with n+1 edges, having nonroot nodes of outdegree 0 or 4. - Emanuele Munarini, Jun 20 2024

Links

Robert Israel, Table of n, a(n) for n = 0..4093

Formula

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

D-finite with recurrence: (-1048576*n^3 - 6291456*n^2 - 11534336*n - 6291456)*a(n) + (1703936*n^3 + 14942208*n^2 + 42336256*n + 38535168)*a(n + 1) + (-589824*n^3 - 5898240*n^2 - 18776064*n - 19169280)*a(n + 2) + (221184*n^3 + 3981312*n^2 + 22978560*n + 42442752)*a(n + 3) + (-18432*n^3 - 1069056*n^2 - 9424896*n - 22487040)*a(n + 4) + (-184320*n^3 - 2810880*n^2 - 14094336*n - 23126016)*a(n + 5) + (69120*n^3 + 1175040*n^2 + 6537600*n + 11827584)*a(n + 6) + (-23328*n^3 - 559872*n^2 - 4430592*n - 11560032)*a(n + 7) + (13608*n^3 + 334368*n^2 + 2731752*n + 7420896)*a(n + 8) + (486*n^3 + 19440*n^2 + 242298*n + 965304)*a(n + 9) + (-729*n^3 - 23328*n^2 - 248751*n - 883872)*a(n + 10) = 0. - Robert Israel, Apr 10 2026

Maple

f:= gfun:-rectoproc({(-1048576*n^3 - 6291456*n^2 - 11534336*n - 6291456)*a(n) + (1703936*n^3 + 14942208*n^2 + 42336256*n + 38535168)*a(n + 1) + (-589824*n^3 - 5898240*n^2 - 18776064*n - 19169280)*a(n + 2) + (221184*n^3 + 3981312*n^2 + 22978560*n + 42442752)*a(n + 3) + (-18432*n^3 - 1069056*n^2 - 9424896*n - 22487040)*a(n + 4) + (-184320*n^3 - 2810880*n^2 - 14094336*n - 23126016)*a(n + 5) + (69120*n^3 + 1175040*n^2 + 6537600*n + 11827584)*a(n + 6) + (-23328*n^3 - 559872*n^2 - 4430592*n - 11560032)*a(n + 7) + (13608*n^3 + 334368*n^2 + 2731752*n + 7420896)*a(n + 8) + (486*n^3 + 19440*n^2 + 242298*n + 965304)*a(n + 9) + (-729*n^3 - 23328*n^2 - 248751*n - 883872)*a(n + 10), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 4, a(7) = 5, a(8) = 10, a(9) = 16}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Apr 10 2026

Prog

(PARI) my(N=50, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^4/(1+x)^3))/x)
(PARI) a(n) = sum(k=0, n\4, (n-4*k+1)*binomial(n+1, k))/(n+1);

Crossrefs

Cf. A001405, A126042.

Cf. A369525, A369581.

Sequence in context: A134220 A179146 A099161 * A033077 A190912 A306108

Adjacent sequences: A369589 A369590 A369591 * A369593 A369594 A369595

Keyword

nonn

Author

Seiichi Manyama, Jan 26 2024

Status

approved