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 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. 4
 1, 3, 72, 2307, 86295, 3513477, 151235361, 6768437853, 311788291023, 14685531568689, 704028657330720, 34239755370728001, 1685178804762196176, 83776625650642935108, 4200738946110487797030, 212201486734654901466543, 10789009182188638106874636, 551682346017956870539952958 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA G.f. A(x) satisfies: (1) 1/A(x^2)^2 + 4*x*A(x^2)^2 = F(x)^4, (2) 1/A(x)^4 + 16*x*A(x)^4 = F(x)^4, (3) 1/A(x^4) + 2*x*A(x^4) = sqrt(F(x)^8 - 4*x), (4) A(x) = ( (F(x)^4 - sqrt(F(x)^8 - 64*x)) / (32*x) )^(1/4), (5) A(x^2) = ( (F(x)^4 - sqrt(F(x)^8 - 16*x)) / (8*x) )^(1/2), where F(x) = (F(x^2)^4 + 8*x)^(1/8) is the g.f. of A223026. EXAMPLE 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 +... PROG (PARI) {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), ", ")) CROSSREFS Cf. A223026. Cf. variants: A187814, A228928. Sequence in context: A332188 A071645 A322189 * A300967 A332721 A332747 Adjacent sequences:  A228709 A228710 A228711 * A228713 A228714 A228715 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 30 2013 STATUS approved

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Last modified September 16 06:18 EDT 2021. Contains 347469 sequences. (Running on oeis4.)