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A349514
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G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - 3 * x).
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1
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1, 4, 24, 192, 1792, 18240, 196224, 2194176, 25247232, 296979456, 3555010560, 43165900800, 530362220544, 6581594275840, 82373440339968, 1038579580796928, 13179023462498304, 168183976239562752, 2157085003249876992, 27790652486543474688, 359485965093121818624
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OFFSET
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0,2
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LINKS
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FORMULA
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a(0) = 1; a(n) = 3 * 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+2*k,3*k) * binomial(3*k,k) * 3^(n-k) / (2*k+1).
a(n) ~ (3/4*(7 + (3*(69 + 16*sqrt(3)))^(1/3) + (3*(69 - 16*sqrt(3)))^(1/3)))^n / (sqrt((4 - (2 + sqrt(3))^(1/3) - (2 - sqrt(3))^(1/3)) * Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 21 2021
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MATHEMATICA
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nmax = 20; A[_] = 0; Do[A[x_] = (1 + x A[x]^3)/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 3 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, 20}]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 20}]
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n+2*k, 3*k) * binomial(3*k, k) * 3^(n-k) / (2*k+1)) \\ Andrew Howroyd, Nov 20 2021
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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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