A158257 -id:A158257 - cp's OEIS frontend

A158258 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} Lucas(n)a(n)x^n/n) where Lucas(n) = A000204(n).

1, 1, 4, 21, 186, 2482, 52431, 1742069, 92198200, 7788221136, 1053871857226, 228795949744458, 79812945269217967, 44781474458725910347, 40447360752560508229164, 58848264986153917140728453

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

			L.g.f.: A(x) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 186*x^5/5 + 2482*x^6/6 +...
exp(A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 44*x^5 + 458*x^6 + 7953*x^7 +...
exp(A(x)) = 1 + x*G(x) where G(x) is the g.f. of A158257 such that:
log(G(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. A158257, A158108 (variant), A000204 (Lucas).

{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,(fibonacci(m-1)+fibonacci(m+1))*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 A158257.

exp(Sum_{n>=1} a(n)*x^n/n) = (1 + Sum_{n>=1} Lucas(n)*a(n)*x^n) / (1 + Sum_{n>=1} (Lucas(n)-1)*a(n)*x^n).

A158109 G.f.: A(x) = exp(Sum_{n>=1} C(2n-1,n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158259(n) and C(2n-1,n) = A001700(n-1).

Original entry on oeis.org

1, 1, 2, 15, 479, 58981, 27087299, 46407723445, 298505825690021, 7255847001783419768, 670260315103084510835973, 236409648316126537191063108559, 319643614642063671478190549232176669

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 479*x^4 + 58981*x^5 +...
log(1+x*A(x)) = x + 1*x^2/2 + 4*x^3/3 + 53*x^4/4 + 2321*x^5/5 +...
log(A(x)) = x + 3*x^2/2 + 40*x^3/3 + 1855*x^4/4 + 292446*x^5/5 +...
log(A(x)) = x + 3*1*x^2/2 + 10*4*x^3/3 + 35*53*x^4/4 + 126*2321*x^5/5 +...

Cf. A158259, A158257 (variant), A001700.

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

cp's OEIS Frontend

A158258 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} Lucas(n)a(n)x^n/n) where Lucas(n) = A000204(n).

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A158109 G.f.: A(x) = exp(Sum_{n>=1} C(2n-1,n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158259(n) and C(2n-1,n) = A001700(n-1).

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cp's OEIS Frontend

A158258 L.g.f.: exp(Sum_{n>=1} a(n)*x^n/n) = 1 + x*exp(Sum_{n>=1} Lucas(n)*a(n)*x^n/n) where Lucas(n) = A000204(n).

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PARI

Formula

A158109 G.f.: A(x) = exp(Sum_{n>=1} C(2n-1,n)*L(n)*x^n/n) such that Sum_{n>=1} L(n)*x^n/n = log(1+x*A(x)) where L(n) = A158259(n) and C(2n-1,n) = A001700(n-1).

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PARI

A158258 L.g.f.: exp(Sum_{n>=1} a(n)x^n/n) = 1 + xexp(Sum_{n>=1} Lucas(n)a(n)x^n/n) where Lucas(n) = A000204(n).

A158109 G.f.: A(x) = exp(Sum_{n>=1} C(2n-1,n)L(n)x^n/n) such that Sum_{n>=1} L(n)x^n/n = log(1+xA(x)) where L(n) = A158259(n) and C(2n-1,n) = A001700(n-1).