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A206724
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E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/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, 3, 10, 65, 326, 3697, 25880, 420177, 4229146, 86320361, 949523972, 27083479909, 352085238818, 11595529550165, 195533892178396, 7357023306710177, 125069396214073010, 5739483201096454633, 109050180820832638028, 5657347053320572351641
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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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E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * exp( Sum_{k>=1} n*x^(n*k)*L(n*k)/(n*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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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 65*x^4/4! + 326*x^5/5! +...
such that the e.g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)*A(-x)/2! + x^3*A(x)*A(u(3)*x)*A(u(3)^2*x)/3! + x^4*A(x)*A(I*x)*A(-x)*A(-I*x)/4! + x^5*A(x)*A(u(5)*x)*A(u(5)^2*x)*A(u(5)^3*x)*A(u(5)^4*x)/5! +...
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 + 4*x^2/2! + 104*x^4/4! + 7332*x^6/6! + 978000*x^8/8! + 213540140*x^10/10! +...
P(3) = 1 + 9*x^3/3! + 5328*x^6/6! + 4805325*x^9/9! + 46498251924*x^12/12! +...
P(4) = 1 + 112*x^4/4! + 1070752*x^8/8! + 84785952288*x^12/12! +...
P(5) = 1 + 75*x^5/5! + 147933500*x^10/10! + 114349984285875*x^15/15! +...
P(6) = 1 + 9036*x^6/6! + 100197544464*x^12/12! + 25022450035686671892*x^18/18! +...
such that A(x) = 1 + x*A(x) + x^2*P(2)/2! + x^3*P(3)/3! + x^4*P(4)/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/m!*A*prod(k=1, m-1, subst(A, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))); polcoeff(round(serlaplace(A)), n)}
for(n=0, 25, 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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