A192768 -id:A192768 - cp's OEIS frontend

A192783 G.f. satisfies: A(x) = Product_{n>=1} 1/(1 - x^n*A(x)^(n^3)).

1, 1, 3, 16, 119, 1145, 13301, 180464, 2821941, 50400230, 1022250876, 23424407915, 602724515761, 17299947151776, 550101222059396, 19246320555772626, 736247255316380311, 30620337253882961105, 1377609185722013042566, 66750666290443384609574

Offset: 0

Paul D. Hanna, Jul 09 2011

nonn

			G.f: A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1145*x^5 + 13301*x^6 +...
The g.f. A = A(x) satisfies:
A = 1/((1 - x*A)*(1 - x^2*A^8)*(1 - x^3*A^27)*(1 - x^4*A^64)*...),
as well as the logarithmic series:
log(A) = x*A + x^2*(A^2 + 2*A^8)/2 + x^3*(A^3 + 3*A^27)/3 + x^4*(A^4 + 2*A^16 + 4*A^64)/4 + x^5*(A^5 + 5*A^125)/5 + x^6*(A^6 + 2*A^24 + 3*A^54 + 6*A^216)/6  +...

Cf. A109085, A192768.

{a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, 1-x^k*(A+x*O(x^n))^(k^3))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x^m/m)*sumdiv(m, d, d*(A+x*O(x^n))^(m*d^2))))); polcoeff(A, n)}

G.f. satisfies: A(x) = exp( Sum_{n>=1} (x^n/n)*Sum_{d|n} d*A(x)^(n*d^2) ).

A192784 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^(n^2)).

Original entry on oeis.org

1, 1, 2, 8, 38, 201, 1164, 7188, 46576, 313823, 2185642, 15668473, 115281167, 868757478, 6696711294, 52757324970, 424590429862, 3490344692094, 29310836035090, 251525003170386, 2206548594680093, 19798923287905907, 181797157100106619

Offset: 0

Paul D. Hanna, Jul 09 2011

nonn

			G.f: A(x) = 1 + x + 2*x^2 + 8*x^3 + 38*x^4 + 201*x^5 + 1164*x^6 +...
The g.f. A = A(x) satisfies:
A = (1 + x*A)*(1 + x^2*A^4)*(1 + x^3*A^9)*(1 + x^4*A^16)*...
as well as the logarithmic series:
log(A) = x*A + x^2*(-A^2 + 2*A^4)/2 + x^3*(A^3 + 3*A^9)/3 + x^4*(-A^4 - 2*A^8 + 4*A^16)/4 + x^5*(A^5 + 5*A^25)/5 + x^6*(-A^6 + 2*A^12 - 3*A^18 + 6*A^36)/6 +...

Cf. A192768.

{a(n)=local(A=1+x); for(i=1, n, A=prod(k=1, n, 1+x^k*A^(k^2)+x*O(x^n))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x^m/m)*sumdiv(m, d, (-1)^(m/d-1)*d*A^(m*d))+x*O(x^n)))); polcoeff(A, n)}

G.f. satisfies: A(x) = exp( Sum_{n>=1} (x^n/n)*Sum_{d|n} -(-1)^(n/d)*d*A(x)^(n*d) ).

cp's OEIS Frontend

A192783 G.f. satisfies: A(x) = Product_{n>=1} 1/(1 - x^n*A(x)^(n^3)).

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A192784 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^(n^2)).

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Author

Keywords

Examples

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Programs

PARI

PARI

Formula