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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
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
Sequence in context: A203154 A097175 A100284 * A341853 A260845 A325157
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.)