A337168 -id:A337168 - cp's OEIS frontend

A337169 a(n) = (-1)^n + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

1, 2, 13, 89, 691, 5720, 49555, 443630, 4071595, 38105342, 362271823, 3488988101, 33967656469, 333752559392, 3305347855573, 32960499084305, 330664662067795, 3335002912108670, 33796042027030855, 343940115478559699, 3513702627928096681, 36021007341027948032

Offset: 0

Ilya Gutkovskiy, Jan 28 2021

nonn

Inverse binomial transform of A005159.

Seiichi Manyama, Table of n, a(n) for n = 0..964

Cf. A000108, A005043, A005159, A337167, A337168.

a[n_] := a[n] = (-1)^n + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 21}]
Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 12], {n, 0, 21}]

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 3*x*A(x)^2.
G.f.: (1 - sqrt(1 - 12*x / (1 + x))) / (6*x).
E.g.f.: exp(5*x) * (BesselI(0,6*x) - BesselI(1,6*x)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * Catalan(k).
a(n) ~ 11^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021

A349255 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 2 * x * A(x)^2)).

Original entry on oeis.org

1, 1, 7, 47, 369, 3113, 27631, 254239, 2403361, 23201393, 227771831, 2266983119, 22822484497, 231994748633, 2377894546783, 24548520253247, 255026759000897, 2664111200687969, 27967731861910759, 294900120348032623, 3121862973452544433, 33167268461833410569

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..950

Cf. A001003, A001764, A153231, A337168, A346627, A349253, A349254, A349256.

nmax = 21; A[] = 0; Do[A[x] = 1/((1 + x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 21}]
a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^3]; Table[a[n], {n, 0, 21}] (* Peter Luschny, Nov 12 2021 *)

a(n) = (-1)^n + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * 2^k * binomial(3*k,k) / (2*k+1).
a(n) = (-1)^n*hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], (3/2)^3). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(171 + 23*sqrt(57)) * (23 + 3*sqrt(57))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(2*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021

A339001 a(n) = (-1)^n * Sum_{k=0..n} (-n)^k * binomial(n,k) * Catalan(k).

Original entry on oeis.org

1, 0, 5, 89, 2481, 93274, 4450645, 258297570, 17689681345, 1397903887808, 125286890408901, 12562851683433765, 1393925069404093105, 169595051215441936902, 22454465186157134883285

Offset: 0

Seiichi Manyama, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..322

Cf. A000108, A005043, A337168, A337169, A338979.

a[0] = 1; a[n_] := (-1)^n * Sum[(-n)^k * Binomial[n, k] * CatalanNumber[k], {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 01 2021 *)

{a(n) = (-1)^n*sum(k=0, n, (-n)^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))}

a(n) = n! * [x^n] exp((2*n-1)*x) * (BesselI(0,2*n*x) - BesselI(1,2*n*x)). - Ilya Gutkovskiy, Feb 01 2021

a(n) ~ exp(-1/4) * 4^n * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 14 2021

cp's OEIS Frontend

A337169 a(n) = (-1)^n + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

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A349255 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 2 * x * A(x)^2)).

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A339001 a(n) = (-1)^n * Sum_{k=0..n} (-n)^k * binomial(n,k) * Catalan(k).

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