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 A337168 a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1). 4
 1, 1, 5, 21, 105, 553, 3053, 17405, 101713, 606033, 3667797, 22485477, 139340985, 871429497, 5492959293, 34862161869, 222592918689, 1428814897825, 9215016141989, 59684122637237, 388045493943049, 2531696701375689, 16569559364596365, 108758426952823709 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Inverse binomial transform of A151374. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 FORMULA G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 2*x*A(x)^2. G.f.: (1 - sqrt(1 - 8*x / (1 + x))) / (4*x). E.g.f.: exp(3*x) * (BesselI(0,4*x) - BesselI(1,4*x)). a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * Catalan(k). a(n) ~ 7^(n + 3/2) / (2^(9/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021 MATHEMATICA a[n_] := a[n] = (-1)^n + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}] Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k CatalanNumber[k], {k, 0, n}], {n, 0, 23}] Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 8], {n, 0, 23}] CROSSREFS Cf. A000108, A005043, A052701, A151374, A162326, A337169. Sequence in context: A203154 A097175 A100284 * A341853 A260845 A325157 Adjacent sequences: A337165 A337166 A337167 * A337169 A337170 A337171 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Jan 28 2021 STATUS approved

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Last modified May 21 17:21 EDT 2024. Contains 372738 sequences. (Running on oeis4.)