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A203318 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) ) where Lucas(n) = A000032(n). 5
1, 1, 2, 4, 9, 16, 36, 64, 135, 250, 504, 917, 1864, 3372, 6593, 12176, 23473, 42732, 82142, 149282, 283104, 516780, 967894, 1757865, 3291964, 5959633, 11039163, 20022457, 36908442, 66637739, 122512809, 220717328, 403499293, 726866565, 1322670966, 2376541137 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Note: 1/(1-x-x^2) = exp(Sum_{n>=1} Lucas(n)*x^n/n) is the g.f. of the Fibonacci numbers.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

G.f.: exp( Sum_{n>=1} A203319(n)*x^n/n ) where A203319(n) = n*fibonacci(n)*Sum_{d|n} 1/(d*fibonacci(d)).

G.f.: exp( Sum_{n>=1} (x^n/n) / (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) ) where Lucas(n) = A000032(n).

G.f.: exp( Sum_{n>=1} F_n(x^n) * x^n/n ) such that F_n(x^n) = Product_{k=0..n-1} F(u^k*x) where F(x) = 1/(1-x-x^2) and u is an n-th root of unity.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...

G.f.: A(x) = exp( Sum_{n>=1} F_n(x^n) * x^n/n )

where F_n(x) = exp( Sum_{k>=1} Lucas(n*k)*x^k/k ), which begin:

F_1(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 13*x^6 + 21*x^7 +...;

F_2(x) = 1 + 3*x + 8*x^2 + 21*x^3 + 55*x^4 + 144*x^5 + 377*x^6 +...;

F_3(x) = 1 + 4*x + 17*x^2 + 72*x^3 + 305*x^4 + 1292*x^5 + 5473*x^6 +...;

F_4(x) = 1 + 7*x + 48*x^2 + 329*x^3 + 2255*x^4 + 15456*x^5 +...;

F_5(x) = 1 + 11*x + 122*x^2 + 1353*x^3 + 15005*x^4 + 166408*x^5 +...;

F_6(x) = 1 + 18*x + 323*x^2 + 5796*x^3 + 104005*x^4 + 1866294*x^5 +...;

...

Also, F_n(x^n) = Product_{k=0..n-1} F(u^k*x) where u = n-th root of unity:

F_1(x) = F(x) = 1/(1-x-x^2) = g.f. of the Fibonacci numbers;

F_2(x^2) = F(x)*F(-x) = 1/(1-3*x^2+x^4);

F_3(x^3) = F(x)*F(w*x)*F(w^2*x) = 1/(1-4*x^3-x^6) where w = exp(2*Pi*I/3);

F_4(x^4) = F(x)*F(I*x)*F(-x)*F(-I*x) = 1/(1-7*x^4+x^8);

F_5(x^5) = 1/(1-11*x^5-x^10);

In general,

F_n(x^n) = 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

...

The logarithmic derivative of this sequence begins:

A203319 = [1,3,7,19,26,81,92,267,358,848,980,3061,3030,7976,...].

PROG

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(m=1, n+1, (x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n)))), n)}

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=local(L=vector(n+1, i, 1)); L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))))); polcoeff(exp(x*Ser(vector(n+1, m, L[m]/m))), n)}

(PARI) {a(n)=local(A=1+x+x*O(x^n), F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); polcoeff(A, n)}

CROSSREFS

Cf. A203319, A203320, A000032 (Lucas); A203413 (Pell variant).

Sequence in context: A128093 A056331 A056336 * A282776 A283015 A264629

Adjacent sequences:  A203315 A203316 A203317 * A203319 A203320 A203321

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 31 2011

STATUS

approved

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Last modified June 15 00:00 EDT 2021. Contains 345041 sequences. (Running on oeis4.)