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Logarithmic derivative of A005169, the number of fountains of n coins.
1

%I #9 Jul 24 2013 15:58:48

%S 1,1,4,5,11,22,36,69,121,221,386,686,1210,2122,3734,6517,11408,19903,

%T 34714,60485,105312,183272,318758,554262,963361,1674076,2908426,

%U 5052066,8774386,15237482,26458718,45939797,79759442,138468656,240382216,417289619,724369536,1257396992

%N Logarithmic derivative of A005169, the number of fountains of n coins.

%F L.g.f.: log( 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(1-...)))))) ), the logarithm of a continued fraction.

%F L.g.f.: log( P(x) / Q(x) ) where

%F P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product(k=1..n} (1-x^k),

%F Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product(k=1..n} (1-x^k),

%F due to the Rogers-Ramanujan continued fraction identity.

%e L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 11*x^5/5 + 22*x^6/6 +...

%e such L(x) = log(P(x)) - log(Q(x)) where

%e P(x) = 1 - x^2 - x^3 - x^4 - x^5 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + x^15 + x^16 - x^18 +...+ A224898(n)*x^n +...

%e Q(x) = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 - 2*x^15 - x^16 - 3*x^17 - 3*x^18 +...+ A039924(n)*x^n +...

%e log(P(x)) = -2*x^2/2 - 3*x^3/3 - 6*x^4/4 - 10*x^5/5 - 11*x^6/6 - 21*x^7/7 - 22*x^8/8 - 39*x^9/9 - 42*x^10/10 +...

%e log(Q(x)) = -x - 3*x^2/2 - 7*x^3/3 - 11*x^4/4 - 21*x^5/5 - 33*x^6/6 - 57*x^7/7 - 91*x^8/8 - 160*x^9/9 - 263*x^10/10 +...

%o (PARI) /* As the log of a continued fraction: */

%o {a(n)=local(A=x, CF=1+x); for(k=0, n, CF=1/(1-x^(n-k+1)*CF+x*O(x^n)); A=log(CF)); n*polcoeff(A, n)}

%o for(n=1,40,print1(a(n),", "))

%o (PARI) /* By the Rogers-Ramanujan continued fraction identity: */

%o {a(n)=local(A=x, P=1+x, Q=1);

%o P=sum(m=0, sqrtint(n), (-1)^m*x^(m*(m+1))/prod(k=1, m, 1-x^k));

%o Q=sum(m=0, sqrtint(n), (-1)^m*x^(m^2)/prod(k=1, m, 1-x^k));

%o A=log(P/(Q+x*O(x^n))); n*polcoeff(A, n)}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A227543, A005169, A039924, A224898.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 17 2013