A168344
G.f. A(x) satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = g.f. of A006664, which is the number of irreducible systems of meanders.
Original entry on oeis.org
1, 1, 3, 15, 99, 773, 6743, 63591, 635307, 6634599, 71759983, 798563065, 9098321475, 105733563393, 1249676348391, 14986826364311, 182027688352427, 2235713532561779, 27732857308708571, 347064951865766607
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 99*x^4 + 773*x^5 + 6743*x^6 +...
A(x) satisfies: A(x*F(x)) = 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 +...
A(x) satisfies: A(x/G(x)) = 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 +...
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F[x_] = (Hypergeometric2F1[-1/2, -1/2, 1, 16x] - 1)/(4x);
A[x_] = x/InverseSeries[x F[x] + O[x]^21, x];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 21 2018, from 2nd formula *)
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{a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)^2));polcoeff(x/serreverse(x*Ser(C_2)),n)}
A168357
Self-convolution of A006664, which is the number of irreducible systems of meanders.
Original entry on oeis.org
1, 2, 5, 20, 112, 768, 5984, 50856, 460180, 4366076, 42988488, 436066232, 4532973676, 48095557700, 519247705968, 5690272928520, 63172884082028, 709373555125356, 8046263496489260, 92089662771965492, 1062482514810065752
Offset: 0
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 +...
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{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)}
A168358
Self-convolution square of A001246, which is the squares of Catalan numbers.
Original entry on oeis.org
1, 2, 9, 58, 458, 4120, 40569, 426842, 4723890, 54402904, 646992474, 7900772120, 98642862232, 1254984808672, 16227116787737, 212790354730842, 2824992774357362, 37915366854924952, 513837166842215970
Offset: 0
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 +...
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Table[Sum[CatalanNumber[k]^2 * CatalanNumber[n-k]^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 10 2018 *)
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{a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)^2)); polcoeff(Ser(C_2)^2, n)}
Showing 1-3 of 3 results.
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