A307413 G.f. A(x) satisfies: A(x) = 1 + x*A(x)/(1 - x*A(x) - 2*x^2*A(x)^2).
1, 1, 2, 7, 26, 102, 420, 1787, 7794, 34666, 156636, 716982, 3317700, 15494156, 72935624, 345701843, 1648489762, 7902956738, 38067806892, 184152092450, 894259126540, 4357738501844, 21302682030328, 104439435098718, 513390992000340, 2529846489669412, 12494572784556440
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Keywords
Programs
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Mathematica
terms = 26; A[] = 0; Do[A[x] = 1 + x A[x]/(1 - x A[x] - 2 x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x] terms = 27; A[] = 0; Do[A[x] = 1 + Sum[(1/3) (2^k - (-1)^k) x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x] terms = 27; CoefficientList[1/x InverseSeries[Series[x (1 + x) (1 - 2 x)/(1 - 2 x^2), {x, 0, terms}], x], x]
Formula
G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Jacobsthal(k)*x^k*A(x)^k, where Jacobsthal = A001045.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 + x)*(1 - 2*x)/(1 - 2*x^2)).
From Paul D. Hanna, Aug 29 2019: (Start)
G.f. A(x) satisfies: 0 = Sum_{n>=1} (1-(-1)^n - 2*A(x))^n * x^n / n.
G.f. A(x) satisfies: log(1 - 4*x^2*A(x)^2)/2 = arctanh(2*x - 2*x*A(x)). (End)
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