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A345105
a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).
1
1, 4, 25, 247, 3283, 54661, 1092427, 25473037, 678837319, 20351864821, 677954261635, 24842157250117, 993040102321927, 43003754679356941, 2005536858420616963, 100211634039201328381, 5341144936822423446247, 302468060262966258380773, 18136282125753572653056355
OFFSET
0,2
FORMULA
E.g.f. A(x) satisfies: A'(x) = 3 * A(x)^2 + exp(x).
MATHEMATICA
a[n_] := a[n] = 1 + 3 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[_] = 1; Do[A[x_] = Normal[Integrate[3 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jun 08 2021
STATUS
approved