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),", "))
A187814
G.f. A(x) satisfies: 1/A(x)^2 + 4*x*A(x)^2 = 1/A(x^2) + 2*x*A(x^2).
Original entry on oeis.org
1, 1, 6, 41, 334, 2901, 26651, 253709, 2483395, 24829132, 252506507, 2603798287, 27161758393, 286118173600, 3039211373800, 32517513415886, 350122302629869, 3790909121211262, 41249405668333107, 450832515809731316, 4947009705400704588, 54479711308604703264, 601933495810972446631
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 41*x^3 + 334*x^4 + 2901*x^5 + 26651*x^6 +...
such that A(x) satisfies the identity illustrated by:
1/A(x)^2 + 4*x*A(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
1/A(x^2) + 2*x*A(x^2) = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
Related expansions.
A(x)^2 = 1 + 2*x + 13*x^2 + 94*x^3 + 786*x^4 + 6962*x^5 + 64793*x^6 +...
A(x)^4 = 1 + 4*x + 30*x^2 + 240*x^3 + 2117*x^4 + 19512*x^5 + 186706*x^6 +...
1/A(x) = 1 - x - 5*x^2 - 30*x^3 - 233*x^4 - 1949*x^5 - 17503*x^6 +...
1/A(x)^2 = 1 - 2*x - 9*x^2 - 50*x^3 - 381*x^4 - 3132*x^5 - 27878*x^6 +...
The g.f. of A107086 begins:
F(x) = 1 + x - x^2 + 2*x^3 - 5*x^4 + 13*x^5 - 35*x^6 + 99*x^7 - 289*x^8 +...
where F(x)^4 = F(x^2)^2 + 4*x:
F(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 + 82*x^7 - 233*x^8 +...
F(x)^4 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 +...
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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*x*subst(A, x, x^2) - 4*x*A^2 +x*O(x^n))^(1/2)); polcoeff(A, n)}
for(n=0, 30, 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 -+...
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{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 1-3 of 3 results.