A219260
G.f. satisfies: A(x) = exp(Sum_{n>=1} x^n*A(x)^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n)).
Original entry on oeis.org
1, 1, 3, 10, 37, 143, 576, 2393, 10178, 44133, 194341, 866867, 3908454, 17784385, 81562890, 376637216, 1749704080, 8171760933, 38346326963, 180707395127, 854850922373, 4057990958069, 19324260613400, 92288612451684, 441919933724974, 2121281845071105, 10205443975074195
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 143*x^5 + 576*x^6 + 2393*x^7 +...
where
log(A(x)) = x*A(x)/1*((1+x*A(x))*(1+x^2*A(x^2))*(1+x^3*A(x^3))*...) +
x^2*A(x)^2/2*((1+x^2*A(x)^2)*(1+x^4*A(x^2)^2)*(1+x^6*A(x^3)^2)*...) +
x^3*A(x)^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(1+x^9*A(x^3)^3)*...) +
x^4*A(x)^4/4*((1+x^4*A(x)^4)*(1+x^8*A(x^2)^4)*(1+x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 101*x^4/4 + 461*x^5/5 + 2144*x^6/6 + 10109*x^7/7 + 48117*x^8/8 + 230998*x^9/9 + 1115875*x^10/10 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A^m, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A219263
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x^n)/n / Product_{k>=1} (1 - x^(n*k)*A(x^n)^k) ).
Original entry on oeis.org
1, 1, 3, 10, 39, 159, 693, 3101, 14292, 67116, 320448, 1549834, 7579037, 37406737, 186102602, 932294987, 4698796087, 23809155711, 121219100012, 619800529792, 3181291257740, 16385813881342, 84666104373097, 438742341955132, 2279628504172080, 11873579440176774, 61984238371422197
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 159*x^5 + 693*x^6 + 3101*x^7 +...
where
log(A(x)) = x*A(x)/1/((1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)*...) +
x^2*A(x^2)/2/((1-x^2*A(x^2))*(1-x^4*A(x^2)^2)*(1-x^6*A(x^2)^3)*...) +
x^3*A(x^3)/3/((1-x^3*A(x^3))*(1-x^6*A(x^3)^2)*(1-x^9*A(x^3)^3)*...) +
x^4*A(x^4)/4/((1-x^4*A(x^4))*(1-x^8*A(x^4)^2)*(1-x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 109*x^4/4 + 531*x^5/5 + 2726*x^6/6 + 13952*x^7/7 + 72581*x^8/8 + 379264*x^9/9 + 1994875*x^10/10 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*subst(A, x, x^m +x*O(x^n))/m/prod(k=1, n\m+1, 1-x^(m*k)*subst(A^k, x, x^m +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-2 of 2 results.
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