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A206725 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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..34.

FORMULA

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 ).

EXAMPLE

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) +...

PROG

(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), ", "))

CROSSREFS

Cf. A206724.

Sequence in context: A029785 A045545 A029790 * A129645 A112994 A056036

Adjacent sequences:  A206722 A206723 A206724 * A206726 A206727 A206728

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 11 2012

STATUS

approved

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Last modified November 27 16:30 EST 2014. Contains 250240 sequences.