A349517 G.f. A(x) satisfies: A(x) = (1 + 4 * x * A(x)^3) / (1 - x).
1, 5, 65, 1145, 23185, 509005, 11782465, 283138545, 6996125985, 176633573205, 4536739406465, 118166489152745, 3113854691067185, 82864654201672605, 2223776891616904065, 60113561634017675745, 1635364503704652830785, 44739382956328846263205, 1230059816693141938275265
Offset: 0
Keywords
Programs
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Mathematica
nmax = 18; A[] = 0; Do[A[x] = (1 + 4 x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] a[0] = 1; a[n_] := a[n] = a[n - 1] + 4 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, 18}] Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 4^k/(2 k + 1), {k, 0, n}], {n, 0, 18}]
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PARI
a(n) = sum(k=0, n, binomial(n+2*k,3*k) * binomial(3*k,k) * 4^k / (2*k+1)) \\ Andrew Howroyd, Nov 20 2021
Formula
a(0) = 1; a(n) = a(n-1) + 4 * 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) * 4^k / (2*k+1).
a(n) ~ sqrt((1 + (1 + 1/phi^(2/3) + phi^(2/3))^3/2) / (2*Pi)) / (6 * n^(3/2) * (1 + 3/phi^(1/3) - 3*phi^(1/3))^n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 21 2021