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A206725
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G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=0..n-1} A(u(n)^k*x), where u(n) = exp(2*Pi*I/n) is an n-th root of unity.
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1
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1, 1, 2, 3, 7, 8, 25, 26, 95, 143, 384, 385, 2030, 2031, 7519, 13114, 45435, 45436, 215811, 215812, 1081077, 1652010, 5091830, 5091831, 31815975, 34705543, 131929631, 218433121, 833547968, 833547969, 4227406719, 4227406720, 22363902821, 33736949254, 109742467470
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies: A(x) = Sum_{n>=0} x^n * exp( Sum_{k>=1} x^(n*k)*L(n*k)/k ) where L(n) is defined by A(x) = exp( Sum_{n>=1} x^n*L(n)/n ).
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 8*x^5 + 25*x^6 + 26*x^7 +...
such that the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)*A(-x) + x^3*A(x)*A(u(3)*x)*A(u(3)^2*x) + x^4*A(x)*A(I*x)*A(-x)*A(-I*x) + x^5*A(x)*A(u(5)*x)*A(u(5)^2*x)*A(u(5)^3*x)*A(u(5)^4*x) +...
where u(n) = exp(2*Pi*I/n) is an n-th root of unity.
The coefficients in P(n) = Product_{k=0..n-1} A(u(n)^k*x) begin:
P(2) = A(x)*A(-x) = 1 + 3*x^2 + 12*x^4 + 53*x^6 + 239*x^8 + 992*x^10 +...
P(3) = 1 + 4*x^3 + 47*x^6 + 270*x^9 + 4912*x^12 + 34610*x^15 +...
P(4) = 1 + 15*x^4 + 304*x^8 + 7989*x^12 + 191611*x^16 + 5101952*x^20 +...
P(5) = 1 + x^5 + 682*x^10 + 21723*x^15 + 2889567*x^20 +...
P(6) = 1 + 78*x^6 + 9873*x^12 + 1213460*x^18 + 195457508*x^24 +...
such that A(x) = 1 + x*A(x) + x^2*P(2) + x^3*P(3) + x^4*P(4) +...
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n+1, x^m*A*round(prod(k=1, m-1, subst(A, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 45, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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