A378483 Expansion of (Sum_{k>=0} binomial(3*k,k) * x^k)^3.
1, 9, 72, 549, 4077, 29772, 214884, 1537677, 10930923, 77298849, 544300992, 3819184236, 26718251868, 186440019192, 1298115301356, 9020928853341, 62582406445287, 433509545320731, 2998884192348888, 20720206275346269, 143005275737941437, 986000187782876976
Offset: 0
Keywords
Programs
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Mathematica
nmax = 20; CoefficientList[Series[Sum[Binomial[3*k,k] * x^k, {k, 0, nmax}]^3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 19 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(3*k, k)*x^k)^3)
Formula
a(n) = Sum_{i+j+k=n, i,j,k >= 0} binomial(3*i,i) * binomial(3*j,j) * binomial(3*k,k).
G.f.: B(x)^3 where B(x) is the g.f. of A005809.
4*a(n) - 27*a(n-1) = 3*A005809(n) for n > 0.
a(n) ~ sqrt(n) * 3^(3*n + 3/2) / (sqrt(Pi) * 2^(2*n+2)). - Vaclav Kotesovec, Jul 19 2025