A223142
G.f. satisfies: A(x)^2 = A(x^2)^2 + 4*x.
Original entry on oeis.org
1, 2, 0, 0, 2, -4, 8, -16, 32, -56, 88, -112, 64, 240, -1264, 4064, -10814, 25500, -54200, 102832, -166020, 190808, 22304, -1058880, 4412424, -13496544, 35306480, -82326496, 172081840, -315115328, 464910368, -363016000, -871587808, 5713552456, -20289991016
Offset: 0
G.f.: A(x) = 1 + 2*x + 2*x^4 - 4*x^5 + 8*x^6 - 16*x^7 + 32*x^8 - 56*x^9 +...
where
A(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 4*x^8 + 4*x^16 + 4*x^32 +...+ 4*x^(2^n) +...
-
{a(n)=local(A=1+x); for(i=1,#binary(n), A=(subst(A, x, x^2)^2+4*x+x*O(x^n))^(1/2)); polcoeff(A, n, x)}
for(n=0,40,print1(a(n),", "))
A223143
G.f. satisfies: A(x)^3 = A(x^2)^3 + 9*x.
Original entry on oeis.org
1, 3, -6, 27, -141, 819, -5022, 31968, -209202, 1398420, -9505854, 65499759, -456410943, 3210397173, -22763553876, 162524220984, -1167359075781, 8429107868541, -61148608627518, 445450238075655, -3257116365714831, 23896262127268719, -175854177039133998
Offset: 0
G.f.: A(x) = 1 + 3*x - 6*x^2 + 27*x^3 - 141*x^4 + 819*x^5 - 5022*x^6 +...
where
A(x)^3 = 1 + 9*x + 9*x^2 + 9*x^4 + 9*x^8 + 9*x^16 + 9*x^32 +...+ 9*x^(2^n) +...
-
{a(n)=local(A=1+x); for(i=1, #binary(n), A=(subst(A, x, x^2)^3+9*x+x*O(x^n))^(1/3)); polcoeff(A, n, x)}
for(n=0,30,print1(a(n),", "))
A228712
G.f. A(x) satisfies: 1/A(x)^4 + 16*x*A(x)^4 = 1/A(x^2)^2 + 4*x*A(x^2)^2.
Original entry on oeis.org
1, 3, 72, 2307, 86295, 3513477, 151235361, 6768437853, 311788291023, 14685531568689, 704028657330720, 34239755370728001, 1685178804762196176, 83776625650642935108, 4200738946110487797030, 212201486734654901466543, 10789009182188638106874636, 551682346017956870539952958
Offset: 0
G.f.: A(x) = 1 + 3*x + 72*x^2 + 2307*x^3 + 86295*x^4 + 3513477*x^5 +...
such that A(x) satisfies the identity illustrated by:
1/A(x)^4 + 16*x*A(x)^4 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 +...
1/A(x^2)^2 + 4*x*A(x^2)^2 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 +...
Related expansions.
A(x)^2 = 1 + 6*x + 153*x^2 + 5046*x^3 + 191616*x^4 + 7876932*x^5 +...
A(x)^4 = 1 + 12*x + 342*x^2 + 11928*x^3 + 467193*x^4 + 19597332*x^5 +...
1/A(x) = 1 - 3*x - 63*x^2 - 1902*x^3 - 69132*x^4 - 2764911*x^5 +...
1/A(x)^2 = 1 - 6*x - 117*x^2 - 3426*x^3 - 122883*x^4 - 4875378*x^5 +...
The g.f. of A223026 begins:
F(x) = 1 + x - 3*x^2 + 14*x^3 - 76*x^4 + 441*x^5 - 2678*x^6 +...
where F(x)^8 = F(x^2)^4 + 8*x:
F(x)^4 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 +...
F(x)^8 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +...
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{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1/(1/subst(A,x,x^2)^2 + 4*x*subst(A,x,x^2)^2 - 16*x*A^4 +x*O(x^n))^(1/4));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
A228711
G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 8*x.
Original entry on oeis.org
1, 2, -5, 22, -115, 646, -3822, 23496, -148368, 955822, -6256273, 41480668, -277954706, 1879118354, -12800031737, 87758481546, -605091552753, 4192829686338, -29180958305391, 203887504096188, -1429568781831693, 10055261467844862, -70929518958227340
Offset: 0
G.f.: A(x) = 1 + 2*x - 5*x^2 + 22*x^3 - 115*x^4 + 646*x^5 - 3822*x^6 +...
where A(x)^4 = A(x^2)^2 + 8*x as demonstrated by:
A(x)^2 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 + 20184*x^7 +...
A(x)^4 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +...
The g.f. of A228712 begins:
G(x) = 1 + 3*x + 72*x^2 + 2307*x^3 + 86295*x^4 + 3513477*x^5 +...
and satisfies: sqrt(1/G(x^2)^2 + 4*x*G(x^2)^2) = A(x).
-
{a(n)=local(A=1+x); for(i=1, n, A=(subst(A, x, x^2)^2+8*x+x*O(x^n))^(1/4)); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))
A228927
G.f. A(x) satisfies: A(x)^16 = A(x^2)^8 + 16*x.
Original entry on oeis.org
1, 1, -7, 70, -798, 9737, -124124, 1631041, -21911758, 299371219, -4144898772, 58007463920, -819038646307, 11650826921489, -166786290656152, 2400680788969898, -34719393978035312, 504223005531434252, -7349846348644213981, 107489242662154350550
Offset: 0
G.f.: A(x) = 1 + x - 7*x^2 + 70*x^3 - 798*x^4 + 9737*x^5 - 124124*x^6 +...
where
A(x)^16 = 1 + 16*x + 8*x^2 - 28*x^4 + 224*x^6 - 2198*x^8 + 23856*x^10 -+...
A(x^2)^8 = 1 + 8*x^2 - 28*x^4 + 224*x^6 - 2198*x^8 + 23856*x^10 -+...
-
{a(n)=local(A=1+x); for(i=1, #binary(n), A=(subst(A, x, x^2)^8+16*x+x*O(x^n))^(1/16)); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))
Showing 1-5 of 5 results.