1, 1, 2, 6, 24, 110, 533, 2673, 13757, 72266, 385940, 2089319, 11439790, 63242238, 352515817, ...

${\displaystyle x(2x^6-5x^5+3x^4-3x^3+6x^2-2x-2)F{\left(x\right)}^3-x(x^6-7x^5+10x^4-6x^3+8x^2-7x-2)F{\left(x\right)}^2-(x^2-2x-1)(2x^4-3x^3+2x^2-3x+1)F\left(x\right)-(x-1)(x^2+1)(x^2-x-1)=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=24}$
${\displaystyle a\left(5\right)=110}$
${\displaystyle a\left(6\right)=533}$
${\displaystyle a\left(7\right)=2673}$
${\displaystyle a\left(8\right)=13757}$
${\displaystyle a\left(9\right)=72266}$
${\displaystyle a\left(10\right)=385940}$
${\displaystyle a\left(11\right)=2089319}$
${\displaystyle a\left(12\right)=11439790}$
${\displaystyle a\left(13\right)=63242238}$
${\displaystyle a\left(14\right)=352515817}$
${\displaystyle a\left(15\right)=1979046864}$
${\displaystyle a\left(16\right)=11180267860}$
${\displaystyle a\left(17\right)=63510804376}$
${\displaystyle a\left(18\right)=362556716475}$
${\displaystyle a\left(19\right)=2078800648342}$
${\displaystyle a\left(20\right)=11966509335141}$
${\displaystyle a\left(21\right)=69131550779070}$
${\displaystyle a\left(22\right)=400680304821635}$
${\displaystyle a\left(23\right)=2329215624325159}$
${\displaystyle a\left(24\right)=13577001634509322}$
${\displaystyle a\left(25\right)=79338848765790520}$
${\displaystyle a\left(26\right)=464699339792308730}$
${\displaystyle a\left(27\right)=2727645599533038445}$
${\displaystyle a\left(28\right)=16042317812182880276}$
${\displaystyle a\left(29\right)=94525723736067575572}$
${\displaystyle a\left(30\right)=557934247481359819342}$
${\displaystyle a\left(31\right)=3298509385545323968801}$
${\displaystyle a\left(32\right)=19530359779784586127068}$
${\displaystyle a\left(33\right)=115803234634851747569433}$
${\displaystyle a\left(34\right)=687563313358980179596077}$
${\displaystyle a\left(35\right)=4087455241866433524477110}$
${\displaystyle a\left(36\right)=24328265974906652484037972}$
${\displaystyle a\left(37\right)=144963653160116788865119184}$
${\displaystyle a\left(38\right)=864711281402797301024675382}$
${\displaystyle a\left(39\right)=5163252287411397456977530156}$
${\displaystyle a\left(40\right)=30859846767054265447391352218}$
${\displaystyle a\left(41\right)=184612869011114357797882139173}$
${\displaystyle a\left(42\right)=1105373408381709459056462285334}$
${\displaystyle a\left(43\right)=6623953254928199696108353311228}$
${\displaystyle a\left(44\right)=39725583356912922419593221620088}$
${\displaystyle a\left(45\right)=238425528754059929098711919185366}$
${\displaystyle a\left(46\right)=1432024146163412587485686431312899}$
${\displaystyle a\left(47\right)=8606956871572391545886008173637700}$
${\displaystyle a\left(48\right)=51765216509953360509430420679528454}$
${\displaystyle a\left(49\right)=311532785426298198235672461809515279}$
${\displaystyle a\left(50\right)=1876012374278334412001617744332431882}$
${\displaystyle a(n+51)=-\frac{n(n+2)a\left(n\right)}{8(2n+103)(n+51)}+\frac{3(5n^2+15n+8)a(n+1)}{32(2n+103)(n+51)}+\frac{(23n^2+187n+300)a(n+2)}{64(2n+103)(n+51)}-\frac{3(73n^2+515n+854)a(n+3)}{64(2n+103)(n+51)}+\frac{(145n^2+1097n+1866)a(n+4)}{64(2n+103)(n+51)}+\frac{3(179n^2+2153n+6228)a(n+5)}{64(2n+103)(n+51)}-\frac{3(247n^2+3791n+13340)a(n+6)}{64(2n+103)(n+51)}-\frac{(997n^2+17747n+79050)a(n+7)}{64(2n+103)(n+51)}+\frac{3(558n^2+12583n+64690)a(n+8)}{32(2n+103)(n+51)}+\frac{(85n^2-5119n-30054)a(n+9)}{64(2n+103)(n+51)}-\frac{(12083n^2+316537n+1988244)a(n+10)}{64(2n+103)(n+51)}+\frac{(6688n^2+197387n+1352133)a(n+11)}{32(2n+103)(n+51)}+\frac{(12295n^2+360533n+2662716)a(n+12)}{64(2n+103)(n+51)}-\frac{(7793n^2+274009n+2274138)a(n+13)}{16(2n+103)(n+51)}+\frac{3(1659n^2+99431n+1046436)a(n+14)}{64(2n+103)(n+51)}+\frac{(10781n^2+441019n+4258431)a(n+15)}{16(2n+103)(n+51)}-\frac{9(3037n^2+168125n+1926284)a(n+16)}{64(2n+103)(n+51)}-\frac{(109609n^2+4388381n+43299096)a(n+17)}{64(2n+103)(n+51)}+\frac{(90257n^2+4094029n+44645250)a(n+18)}{32(2n+103)(n+51)}+\frac{3(2986n^2+98846n+880347)a(n+19)}{32(2n+103)(n+51)}-\frac{(226889n^2+11515669n+140767950)a(n+20)}{64(2n+103)(n+51)}+\frac{(172919n^2+9639985n+126774906)a(n+21)}{64(2n+103)(n+51)}+\frac{(4309n^2+1295963n+28147590)a(n+22)}{64(2n+103)(n+51)}+\frac{(260n^2-1068371n-25544958)a(n+23)}{16(2n+103)(n+51)}-\frac{3(66519n^2+2930639n+31859810)a(n+24)}{32(2n+103)(n+51)}+\frac{3(95073n^2+5570874n+80157814)a(n+25)}{32(2n+103)(n+51)}+\frac{3(20473n^2+182610n-9298656)a(n+26)}{32(2n+103)(n+51)}-\frac{(964085n^2+55422007n+791865954)a(n+27)}{64(2n+103)(n+51)}+\frac{(618050n^2+39390319n+614370576)a(n+28)}{32(2n+103)(n+51)}-\frac{(434522n^2+25404124n+362341623)a(n+29)}{32(2n+103)(n+51)}+\frac{(720095n^2+29049727n+203437320)a(n+30)}{64(2n+103)(n+51)}-\frac{(1405165n^2+72878183n+901291428)a(n+31)}{64(2n+103)(n+51)}+\frac{3(217771n^2+13983175n+227562428)a(n+32)}{64(2n+103)(n+51)}+\frac{(817067n^2+47290723n+663554133)a(n+33)}{32(2n+103)(n+51)}-\frac{3(383128n^2+24895513n+403550962)a(n+34)}{32(2n+103)(n+51)}+\frac{(462367n^2+35863043n+693680724)a(n+35)}{16(2n+103)(n+51)}-\frac{(1602805n^2+137276159n+2885843160)a(n+36)}{64(2n+103)(n+51)}+\frac{(856475n^2+68127418n+1349007570)a(n+37)}{32(2n+103)(n+51)}-\frac{3(442612n^2+32140590n+578076831)a(n+38)}{32(2n+103)(n+51)}+\frac{3(309147n^2+21192810n+350625457)a(n+39)}{32(2n+103)(n+51)}+\frac{(629567n^2+55759939n+1232986278)a(n+40)}{32(2n+103)(n+51)}-\frac{(702529n^2+58407083n+1214979753)a(n+41)}{16(2n+103)(n+51)}+\frac{3(165957n^2+14172356n+302625615)a(n+42)}{16(2n+103)(n+51)}-\frac{(158792n^2+14301775n+321741069)a(n+43)}{16(2n+103)(n+51)}-\frac{(134461n^2+11362763n+239290914)a(n+44)}{32(2n+103)(n+51)}+\frac{3(13270n^2+1186021n+26485254)a(n+45)}{8(2n+103)(n+51)}-\frac{3(1080n^2+98293n+2230546)a(n+46)}{8(2n+103)(n+51)}-\frac{(9055n^2+839531n+19455456)a(n+47)}{16(2n+103)(n+51)}-\frac{(53n^2+6886n+209304)a(n+48)}{4(2n+103)(n+51)}+\frac{(160n^2+16004n+400209)a(n+49)}{4(2n+103)(n+51)}+\frac{(34n^2+3431n+86553)a(n+50)}{4(2n+103)(n+51)},n\ge 51}$