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 A251180 G.f. satisfies: A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n / (1+2*x)^(n+1). 4
 1, 1, 2, 11, 59, 376, 2566, 18646, 141857, 1120851, 9141387, 76635239, 658411100, 5784858465, 51899580702, 474971067333, 4431203311040, 42128438013171, 408111843546201, 4028707682556147, 40534978365189110, 415825232653264747, 4350847058443120856, 46450772334813948748 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare to: F(x) = Sum_{n>=0} (F(x)^n + 2)^n * x^n / (1+3*x)^(n+1) holds when F(x) = 1. LINKS FORMULA G.f. satisfies: (1) A(x) = Sum_{n>=0} A(x)^(n^2) * x^n / (1+2*x - 2*x*A(x)^n)^(n+1). (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * (A(x)^k + 2)^k. EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 59*x^4 + 376*x^5 + 2566*x^6 +... where we have the identities: (0) A(x) = 1/(1+2*x) + (A(x)+2)*x/(1+2*x)^2 + (A(x)^2+2)^2*x^2/(1+2*x)^3 + (A(x)^3+2)^3*x^3/(1+2*x)^4 + (A(x)^4+2)^4*x^4/(1+2*x)^5 + (A(x)^5+2)^5*x^5/(1+2*x)^6 +... (1) A(x) = 1 + A(x)*x/(1+2*x - 2*x*A(x))^2 + A(x)^4*x^2/(1+2*x - 2*x*A(x)^2)^3 + A(x)^9*x^3/(1+2*x - 2*x*A(x)^3)^4 + A(x)^16*x^4/(1+2*x - 2*x*A(x)^4)^5 + A(x)^25*x^5/(1+2*x - 2*x*A(x)^5)^6 + A(x)^36*x^6/(1+2*x - 2*x*A(x)^6)^7 +... (2) A(x) = 1 - x*(2 - (A(x)+2)) + x^2*(2^2 - 2*2*(A(x)+2) + (A(x)^2+2)^2) - x^3*(2^3 - 3*2^2*(A(x)+2) + 3*2*(A(x)^2+2)^2 - (A(x)^3+2)^3) + x^4*(2^4 - 4*2^3*(A(x)+2) + 6*2^2*(A(x)^2+2)^2 - 4*2*(A(x)^3+2)^3 + (A(x)^4+2)^4) - x^5*(2^5 - 5*2^4*(A(x)+2) + 10*2^3*(A(x)^2+2)^2 - 10*2^2*(A(x)^3+2)^3 + 5*2*(A(x)^4+2)^4 - (A(x)^5+2)^5) +... PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (A^m + 2)^m * x^m / (1+2*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, A^(m^2) * x^m / (1+2*x - 2*x*A^m +x*O(x^n) )^(m+1) )); polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m * sum(k=0, m, binomial(m, k) * (-2)^(m-k) * (A^k + 2)^k +x*O(x^n)))); polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A251177, A251178, A251179, A251181, A244610. Sequence in context: A139172 A228868 A290116 * A286194 A164034 A240548 Adjacent sequences:  A251177 A251178 A251179 * A251181 A251182 A251183 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 19 2015 STATUS approved

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)