A372019 G.f. A(x) satisfies A(x) = ( 1 + 9*x*A(x)/(1 - x*A(x)) )^(1/3).
1, 3, 3, 3, 30, 57, 84, 867, 1893, 3162, 33132, 76953, 136812, 1446204, 3478764, 6420387, 68260134, 167946159, 317782524, 3392340186, 8479140510, 16332164868, 174873206424, 442212416121, 863222622780, 9264327739716, 23637757714788, 46624054987452
Offset: 0
Keywords
Programs
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Maple
A371019 := proc(n) add(9^k*binomial((n+1)/3,k)*binomial(n-1,k-1),k=0..n) ; %/(n+1) ; end proc: seq(A371019(n),n=0..60) ; # R. J. Mathar, Apr 22 2024
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PARI
a(n) = sum(k=0, n, 9^k*binomial(n/3+1/3, k)*binomial(n-1, n-k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} 9^k * binomial(n/3+1/3,k) * binomial(n-1,n-k).
D-finite with recurrence n*(n-1)*(n+1)*a(n) -8*(2*n-5)*(8*n^2-40*n+57)*a(n-3) +4096*(n-5)*(n-6)*(n-4)*a(n-6)=0. - R. J. Mathar, Apr 22 2024