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%I #64 Aug 08 2023 07:57:52
%S 1,1,5,34,267,2279,20540,192350,1853255,18252079,182924645,1859546968,
%T 19127944500,198725331588,2082256791048,21979169545670,
%U 233495834018591,2494624746580655,26786319835972799,288915128642169250,3128814683222599331,34007373443388857999
%N Number of Sylvester classes of 3-packed words of degree n.
%C See Novelli-Thibon (2014) for precise definition.
%H Seiichi Manyama, <a href="/A243659/b243659.txt">Table of n, a(n) for n = 0..941</a> (terms 1..100 from Lars Blomberg)
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Eq. (185), p. 47 and Fig. 17.
%F Novelli-Thibon give an explicit formula in Eq. (182).
%F a(0) = 1 and a(n) = (1/n) * Sum_{i=1..n} ( Sum_{j=0..3*i} (2^(j-2*i)*(-1)^(i-j) * binomial(i,3*i-j)*binomial(i+j-1,i-1)) *a(n-i) ) for n > 0. - _Vladimir Kruchinin_, Apr 09 2017
%F From _Seiichi Manyama_, Jul 26 2020: (Start)
%F G.f. A(x) satisfies: A(x) = 1 - x * A(x)^3 * (1 - 2 * A(x)).
%F a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1).
%F a(n) = ( (-1)^n / (3*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k) * binomial(4*n-k,n-k). (End)
%F a(n) ~ sqrt(24388 + 9221*sqrt(7)) * (316 + 119*sqrt(7))^(n - 1/2) / (sqrt(7*Pi) * n^(3/2) * 2^(n + 3/2) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Jul 31 2021
%F a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0. - _Seiichi Manyama_, Aug 08 2023
%t b[0] = 1; b[n_] := b[n] = 1/n Sum[Sum[2^(j-2i)(-1)^(i-j) Binomial[i, 3i-j] Binomial[i+j-1, i-1], {j, 0, 3i}] b[n-i], {i, 1, n}];
%t a[n_] := b[n+1];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 27 2018, after _Vladimir Kruchinin_ *)
%o (Maxima)
%o a(n):=if n=0 then 1 else 1/n*sum(sum(2^(j-2*i)*(-1)^(i-j)*binomial(i,3*i-j)*binomial(i+j-1,i-1),j,0,3*i)*a(n-i),i,1,n); /* _Vladimir Kruchinin_, Apr 07 2017 */
%o (PARI) a(n) = if(n==0, 1, sum(i=1, n, a(n-i)*sum(j=0, 3*i, 2^(j-2*i)*(-1)^(i-j)*binomial(i,3*i-j)*binomial(i+j-1,i-1)))/n); \\ _Seiichi Manyama_, Jul 26 2020
%o (PARI) a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^3*(1-2*A)); polcoeff(A, n); \\ _Seiichi Manyama_, Jul 26 2020
%o (PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1)); \\ _Seiichi Manyama_, Jul 26 2020
%o (PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(3*n+1, k)*binomial(4*n-k, n-k))/(3*n+1); \\ _Seiichi Manyama_, Jul 26 2020
%Y Column k=3 of A336573.
%Y Cf. A003168, A336539.
%Y Cf. A002293, A349331, A364747, A364765.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 12 2014
%E a(9)-a(21) from _Lars Blomberg_, Jul 12 2017
%E a(0)=1 inserted by _Seiichi Manyama_, Jul 26 2020
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