A158495
Expansion of ((1-4*x) + sqrt(1-4*x))/(2*(1-2*x)).
Original entry on oeis.org
1, -1, -3, -8, -21, -56, -154, -440, -1309, -4048, -12958, -42712, -144210, -496432, -1735764, -6145968, -21986781, -79331232, -288307254, -1054253208, -3875769606, -14315659632, -53097586284, -197677736208, -738415086066
Offset: 0
Essentially the same as
A014318, up to sign and offset.
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A158495:= func< n | n eq 0 select 1 else - (&+[2^(n-j)*Catalan(j-1): j in [1..n]]) >;
[A158495(n): n in [0..40]]; // G. C. Greubel, Jan 09 2023
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CoefficientList[Series[((1-4x)+Sqrt[1-4x])/(2(1-2x)),{x,0,30}],x] (* Harvey P. Dale, Dec 15 2011 *)
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def A158495(n): return int(n==0) - sum(2^(n-k)*catalan_number(k-1) for k in range(1,n+1))
[A158495(n) for n in range(41)] # G. C. Greubel, Jan 09 2023
A307489
G.f. A(x) satisfies: A(x) = 1/(1 - 2*x*A(x) - x*A(x)/(1 - x*A(x)/(1 - x*A(x)/(1 - ...)))), a continued fraction.
Original entry on oeis.org
1, 3, 19, 152, 1367, 13195, 133556, 1398696, 15029311, 164764985, 1835614027, 20722066612, 236524088612, 2725081792932, 31649837891768, 370161223462480, 4355751419996559, 51532214460643957, 612604251998847641, 7313900470316335280, 87659840436181657215
Offset: 0
G.f.: A(x) = 1 + 3*x + 19*x^2 + 152*x^3 + 1367*x^4 + 13195*x^5 + 133556*x^6 + 1398696*x^7 + 15029311*x^8 + 164764985*x^9 + 1835614027*x^10 + ...
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terms = 21; CoefficientList[1/x InverseSeries[Series[x (1 - 4 x + Sqrt[1 - 4 x])/2, {x, 0, terms}], x], x]
terms = 21; A[] = 0; Do[A[x] = 1/(1 - 2 x A[x] + ContinuedFractionK[-x A[x], 1, {k, 1, j}]) + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 20; A[] = 1; Do[A[x] = 2/(1 - 4 x A[x] + Sqrt[1 - 4 x A[x]]) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 21; A[] = 1; Do[A[x] = Sum[Binomial[2 k + 1, k + 1] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
A308850
Expansion of e.g.f. exp(-2*x) / (BesselI(0,2*x) + BesselI(1,2*x)).
Original entry on oeis.org
1, -3, 8, -17, 18, 58, -364, 369, 6194, -37382, -28848, 1717274, -8592644, -47472804, 918146560, -2911313551, -61122074382, 806675821162, 46813084592, -105331573943466, 1018198168087636, 6417696715221572, -247555432672498872, 1535509971584425358, 34028097257000628028, -764203552200012087252
Offset: 0
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nmax = 25; CoefficientList[Series[Exp[-2 x]/(BesselI[0, 2 x] + BesselI[1, 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] Binomial[2 k + 1, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
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