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A349318
G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 2 * x).
2
1, 1, 5, 28, 171, 1113, 7590, 53588, 388519, 2876003, 21648065, 165193576, 1275043280, 9936953788, 78087083456, 618049278976, 4922606097263, 39425205882007, 317316076325015, 2565216211152700, 20819872339143179, 169586043613302169, 1385856599443533442
OFFSET
0,3
FORMULA
a(0) = a(1) = 1; a(n) = 2 * a(n-1) + 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} binomial(n-1,k-1) * binomial(3*k,k) * 2^(n-k) / (2*k+1).
a(n) ~ 35^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Nov 25 2021
MATHEMATICA
nmax = 22; A[_] = 0; Do[A[x_] = 1 + x A[x]^3/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + 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, 22}]
Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 21 2021
STATUS
approved