A158258 -id:A158258 - cp's OEIS frontend

A158257 G.f.: A(x) = exp(Sum_{n>=1} Lucas(n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158258(n) and Lucas(n) = A000204(n).

1, 1, 2, 7, 44, 458, 7953, 225761, 10470604, 789302962, 96596105976, 19162936947418, 6158621106553275, 3204835468356347519, 2699695571885775547222, 3680716263445262350996413

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 44*x^4 + 458*x^5 + 7953*x^6 +...
log(1+x*A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 186*x^5/5 + 2482*x^6/6 +...
log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 147*x^4/4 + 2046*x^5/5 + 44676*x^6/6 +...
log(A(x)) = x + 3*1*x^2/2 + 4*4*x^3/3 + 7*21*x^4/4 + 11*186*x^5/5 + 18*2482*x^6/6 +...

Cf. A158258, A158107 (variant), A000204 (Lucas).

{a(n)=local(A=1+x);if(n==0,1,for(i=1,n,A=exp(sum(m=1,n,(fibonacci(m-1)+fibonacci(m+1))*x^m*polcoeff(log(1+x*A+x*O(x^m)),m))+x*O(x^n)));polcoeff(A,n))}

A158259 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} C(2n-1,n)a(n)x^n/n) where C(2n-1,n) = A001700(n-1).

Original entry on oeis.org

1, 1, 4, 53, 2321, 351010, 189198136, 371045084781, 2686134761118382, 72555484959298332681, 7372783651816395650943931, 2836907736669733620359204710274, 4155363917021399525198623243750199333

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

			L.g.f.: A(x) = x + 1*x^2/2 + 4*x^3/3 + 53*x^4/4 + 2321*x^5/5 +...
exp(A(x)) = 1 + x + 2*x^2 + 15*x^3 + 479*x^4 + 58981*x^5 +...
exp(A(x)) = 1 + x*G(x) where G(x) is the g.f. of A158109 such that:
log(G(x)) = x + 3*1*x^2/2 + 10*4*x^3/3 + 35*53*x^4/4 + 126*2321*x^5/5 +...

Cf. A158109, A158258 (variant), A001700.

{a(n)=local(A=x+x^2);if(n==0,1,for(i=1,n-1,A=log(1+x*exp(sum(m=1,n,binomial(2*m-1,m)*x^m*polcoeff(A+x*O(x^m),m) )+x*O(x^n))));n*polcoeff(A,n))}

L.g.f.: exp(Sum_{n>=1} a(n)*x^n/n) = 1 + x*G(x) where G(x) = g.f. of A158109.

exp(Sum_{n>=1} a(n)*x^n/n) = [1 + Sum_{n>=1} C(2n-1,n)*a(n)*x^n]/[1 + Sum_{n>=1} (C(2n-1,n)-1)*a(n)*x^n].

cp's OEIS Frontend

A158257 G.f.: A(x) = exp(Sum_{n>=1} Lucas(n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158258(n) and Lucas(n) = A000204(n).

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A158259 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} C(2n-1,n)a(n)x^n/n) where C(2n-1,n) = A001700(n-1).

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cp's OEIS Frontend

A158257 G.f.: A(x) = exp(Sum_{n>=1} Lucas(n)*L(n)*x^n/n) such that Sum_{n>=1} L(n)*x^n/n = log(1+x*A(x)) where L(n) = A158258(n) and Lucas(n) = A000204(n).

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PARI

A158259 L.g.f.: exp(Sum_{n>=1} a(n)*x^n/n) = 1 + x*exp(Sum_{n>=1} C(2n-1,n)*a(n)*x^n/n) where C(2n-1,n) = A001700(n-1).

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PARI

Formula

A158257 G.f.: A(x) = exp(Sum_{n>=1} Lucas(n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158258(n) and Lucas(n) = A000204(n).

A158259 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} C(2n-1,n)a(n)x^n/n) where C(2n-1,n) = A001700(n-1).