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Number of n-strand braids of length at most 2 in the dual monoid B_n^{+*}.
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5
1, 4, 83, 556, 11124, 266944
Number of n-strand braids of length at most 6 in the dual monoid B_n^{+*}.
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5
G.f. A(x) satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = g.f. of A004304, where A004304(n) is the number of planar tree-rooted maps with n edges.
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4
1, 2, 6, 26, 148, 1012, 7824, 65886, 590452, 5546972, 54070432, 542937320, 5586265280, 58659600352, 626702981084, 6795682231830, 74645847739012, 829257675740724, 9304974123394272, 105343378754088424
FORMULA
G.f.: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1). G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A005568. G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) = g.f. of A004304.
EXAMPLE
G.f. A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 148*x^4 + 1012*x^5 + 7824*x^6 +...
A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A005568: F(x) = 1 + 2*x + 10*x^2 + 70*x^3 + 588*x^4 + 5544*x^5 + 56628*x^6 +...+ A000108(n)* A000108(n+1)*x^n +... A(x) satisfies: A(x/G(x)) = G(x) = g.f. of A004304: G(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 28*x^4 + 160*x^5 + 1036*x^6 +...
PROG
(PARI) {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)*(binomial(2*m, m)/(m+1)))); polcoeff((x/serreverse(x*Ser(C_2))), n)}
Self-convolution of A006664, which is the number of irreducible systems of meanders.
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3
1, 2, 5, 20, 112, 768, 5984, 50856, 460180, 4366076, 42988488, 436066232, 4532973676, 48095557700, 519247705968, 5690272928520, 63172884082028, 709373555125356, 8046263496489260, 92089662771965492, 1062482514810065752
FORMULA
G.f.: A(x) = x/Series_Reversion(x*F(x)^2) where F(x) = g.f. of A001246, which is the squares of Catalan numbers. G.f.: A(x) = F(x/A(x))^2 where A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A001246.
EXAMPLE
G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 112*x^4 + 768*x^5 +...
A(x)^(1/2) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...+ A006664(n)*x^n +... G.f. satisfies: A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A001246: F(x) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A000108(n)^2*x^n +... F(x)^2 = 1 + 2*x + 9*x^2 + 58*x^3 + 458*x^4 + 4120*x^5 + 40569*x^6 +...+ A168358(n)*x^n +...
PROG
(PARI) {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)^2)); polcoeff(x/serreverse(x*Ser(C_2)^2), n)}
Self-convolution square of A001246, which is the squares of Catalan numbers.
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2
1, 2, 9, 58, 458, 4120, 40569, 426842, 4723890, 54402904, 646992474, 7900772120, 98642862232, 1254984808672, 16227116787737, 212790354730842, 2824992774357362, 37915366854924952, 513837166842215970
FORMULA
G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)^2) where G(x) = g.f. of A006664, which is the number of irreducible systems of meanders. G.f.: A(x) = G(x*A(x))^2 where A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A006664. Recurrence: (n+1)^2*(n+2)^3*(4*n^2 - 5*n - 3)*a(n) = 4*(n+1)^2*(48*n^5 - 12*n^4 - 136*n^3 + 15*n^2 + 49*n - 30)*a(n-1) - 32*(96*n^7 - 312*n^6 + 104*n^5 + 580*n^4 - 630*n^3 + 80*n^2 + 91*n - 12)*a(n-2) + 1024*(n-2)^3*(2*n - 3)^2*(4*n^2 + 3*n - 4)*a(n-3).
a(n) ~ (4/Pi - 1) * 2^(4*n + 3) / (Pi*n^3). (End)
EXAMPLE
G.f.: A(x) = 1 + 2*x + 9*x^2 + 58*x^3 + 458*x^4 + 4120*x^5 +...
A(x)^(1/2) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A001246(n)*x^n +... A(x) satisfies: A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A006664: G(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...+ A006664(n)*x^n +... G(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 112*x^4 + 768*x^5 + 5984*x^6 +...+ A168357(n)*x^n +...
MATHEMATICA
Table[Sum[CatalanNumber[k]^2 * CatalanNumber[n-k]^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 10 2018 *)
PROG
(PARI) {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)^2)); polcoeff(Ser(C_2)^2, n)}
G.f. A(x) satisfies: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A133053, which is the squares of Motzkin numbers ( A001006).
+10
0
1, 1, 3, 6, 20, 70, 302, 1386, 6902, 35862, 194202, 1082642, 6191680, 36141118, 214715244, 1294849186, 7911159522, 48888093910, 305165808290, 1921992409066, 12202404037088, 78031629139246, 502263432618224, 3252160882871210
FORMULA
G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A133053.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 20*x^4 + 70*x^5 + 302*x^6 +...
A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A133053: F(x) = 1 + x + 4*x^2 + 16*x^3 + 81*x^4 + 441*x^5 + 2601*x^6 +...+ A001006(n)^2*x^n +...
PROG
(PARI) {a(n)=if(n==0, 1, polcoeff(x/serreverse(x*sum(m=0, n, polcoeff((1/x)*serreverse(x/(1+x+x^2+x^2*O(x^m))), m)^2 *x^m)+x^2*O(x^n)), n))}
Number of n-strand braids of length at most 3 in the dual monoid B_n^{+*}.
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0
1, 5, 177, 2856, 147855, 9845829
Number of n-strand braids of length at most 4 in the dual monoid B_n^{+*}.
+10
0
1, 6, 367, 14122, 1917046, 356470124
Number of n-strand braids of length at most 5 in the dual monoid B_n^{+*}.
+10
0
1, 7, 749, 68927, 24672817
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