A107086
G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 4*x.
Original entry on oeis.org
1, 1, -1, 2, -5, 13, -35, 99, -289, 857, -2578, 7864, -24252, 75430, -236348, 745431, -2364399, 7536482, -24127482, 77544613, -250098478, 809169322, -2625483810, 8541037140, -27851360659, 91018956200, -298052119611, 977825373366, -3213513271929, 10577811289462, -34870732260397
Offset: 0
A(x)^4 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
A(x^2)^2 = 1 + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
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nmin = 0; nmax = 30; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^4 - A[x^2]^2 - 4x + O[x]^(n+1), x][[2;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 2, nmax}];
a /@ Range[nmin, nmax] /. sol (* Jean-François Alcover, Nov 07 2019 *)
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{a(n)=local(A=1+x); for(i=1, n, A=(subst(A, x, x^2)^2+4*x+x*O(x^n))^(1/4)); polcoeff(A, n, x)}
for(n=0,40,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),", "))
A228928
G.f. A(x) satisfies: 1/A(x)^8 + 64*x*A(x)^8 = 1/A(x^2)^4 + 8*x*A(x^2)^4.
Original entry on oeis.org
1, 7, 672, 91147, 14486409, 2516759469, 463051052653, 88674496050245, 17490154693966234, 3528922457876864195, 724934544034900295558, 151110852750623222310189, 31881833636363854856989129, 6795336519252277650628254056, 1461001691259055273207790036665
Offset: 0
G.f.: A(x) = 1 + 7*x + 672*x^2 + 91147*x^3 + 14486409*x^4 +...
such that A(x) satisfies the identity illustrated by:
1/A(x)^8 + 64*x*A(x)^8 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 +...
1/A(x^2)^4 + 8*x*A(x^2)^4 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 +...
Related expansions.
A(x)^4 = 1 + 28*x + 2982*x^2 + 422408*x^3 + 68709025*x^4 + 12111355116*x^5 +...
A(x)^8 = 1 + 56*x + 6748*x^2 + 1011808*x^3 + 169965222*x^4 + 30589656944*x^5 +...
1/A(x)^4 = 1 - 28*x - 2198*x^2 - 277368*x^3 - 42560861*x^4 - 7240234148*x^5 +...
1/A(x)^8 = 1 - 56*x - 3612*x^2 - 431648*x^3 - 64757910*x^4 - 10877750352*x^5 +...
The g.f. of A228927 satisfies F(x) = (F(x^2)^8 + 16*x)^(1/16) and begins:
F(x) = 1 + x - 7*x^2 + 70*x^3 - 798*x^4 + 9737*x^5 - 124124*x^6 + 1631041*x^7 +...
where F(x)^16 = F(x^2)^8 + 16*x:
F(x)^8 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 - 277368*x^6 +...
F(x)^16 = 1 + 16*x + 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+x*O(x^n)); for(i=1, n, A=1/(1/subst(A, x, x^2)^4 + 8*x*subst(A, x, x^2)^4 - 64*x*A^8 +x*O(x^n))^(1/8)); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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