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 A323574 G.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * ((1+x)^n - 1)^n / (A(x) + 2 - 2*(1+x)^n)^(n+1). 1
 1, 2, 20, 320, 7996, 269272, 11293792, 563253696, 32433009160, 2113347523336, 153579286783456, 12309659862402976, 1078628781953636960, 102578628758305245024, 10523148808846566898816, 1158407291029244188955264, 136214299772837816557703120, 17040721610970237566148646464, 2260018461602151565512432884608, 316748455363386162460484685488512 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA G.f. A(x) satisfies: (1) 1 = Sum_{n>=0} 2^n * ((1+x)^n - 1)^n/(A(x) + 2 - 2*(1+x)^n)^(n+1). (2) 1 = Sum_{n>=0} 2^n * ((1+x)^n + 1)^n/(A(x) + 2 + 2*(1+x)^n)^(n+1). EXAMPLE G.f.: A(x) = 1 + 2*x + 20*x^2 + 320*x^3 + 7996*x^4 + 269272*x^5 + 11293792*x^6 + 563253696*x^7 + 32433009160*x^8 + 2113347523336*x^9 + 153579286783456*x^10 + ... such that 1 = 1/A(x) + 2*((1+x) - 1)/(A(x) + 2 - 2*(1+x))^2 + 2^2*((1+x)^2 - 1)^2/(A(x) + 2 - 2*(1+x)^2)^3 + 2^3*((1+x)^3 - 1)^3/(A(x) + 2 - 2*(1+x)^3)^4 + 2^4*((1+x)^4 - 1)^4/(A(x) + 2 - 2*(1+x)^4)^5 + 2^5*((1+x)^5 - 1)^5/(A(x) + 2 - 2*(1+x)^5)^6 + ... also, 1 = 1/(A(x) + 4) + 2*(1 + (1+x))/(A(x) + 2 + 2*(1+x))^2 + 2^2*(1 + (1+x)^2)^2/(A(x) + 2 + 2*(1+x)^2)^3 + 2^3*(1 + (1+x)^3)^3/(A(x) + 2 + 2*(1+x)^3)^4 + 2^4*(1 + (1+x)^4)^4/(A(x) + 2 + 2*(1+x)^4)^5 + 2^5*(1 + (1+x)^5)^5/(A(x) + 2 + 2*(1+x)^5)^6 + ... PROG (PARI) {a(n) = my(A=[1], X=x+x*O(x^n)); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 2^m * ((1+X)^m - 1)^m / (Ser(A) + 2 - 2*(1+X)^m)^(m+1) ) )[#A]); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A323313. Sequence in context: A367862 A177397 A360342 * A294454 A322729 A304861 Adjacent sequences: A323571 A323572 A323573 * A323575 A323576 A323577 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 06 2019 STATUS approved

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Last modified July 24 10:37 EDT 2024. Contains 374583 sequences. (Running on oeis4.)