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A346627
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G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * A(x)^3.
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9
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1, 0, 1, 2, 7, 23, 82, 300, 1129, 4334, 16914, 66899, 267586, 1080516, 4398850, 18035084, 74402361, 308624282, 1286428765, 5385578256, 22635057148, 95471113565, 403983783772, 1714494024947, 7295949019114, 31124885587680, 133085594104222, 570266646942488
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OFFSET
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0,4
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COMMENTS
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Inverse binomial transform of A200753.
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LINKS
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FORMULA
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G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 + x)^(2*k+1).
a(n) = (-1)^n + 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) * binomial(3*k,k) / (2*k + 1).
a(n) ~ sqrt(198 + 38*sqrt(33)) * (19 + 3*sqrt(33))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 3)). - Vaclav Kotesovec, Jul 30 2021
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MATHEMATICA
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nmax = 27; A[_] = 0; Do[A[x_] = 1/(1 + x) + x A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 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, 27}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 27}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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