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A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A003701(n)*x^n. 0

%I #5 Mar 30 2012 18:37:21

%S 1,1,1,0,3,0,38,0,947,0,37394,0,2120190,0,162980012,0,16330173251,0,

%T 2070201641498,0,324240251016266,0,61525045423103316,0,

%U 13913915097436287598,0,3698477457114061621492,0

%N G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A003701(n)*x^n.

%C The e.g.f. of A003701 is exp(x)/cos(x) = Sum_{n>=0} A003701(n)*x^n/n!.

%C Compare to A157308 and A157310.

%F a(n) = |A157308(n)| = |A157310(n)| for n>=0.

%F a(2n) = A158119(n) for n>=0; a(2n-1) = 0 for n>=2, with a(1)=1.

%F G.f. A = A(x) satisfies: A(x) = 1/(1-x/A - (x/A)^2/(1-x/A - 2^2*(x/A)^2/(1-x/A - 3^2*(x/A)^2/(1-x/A - 4^2*(x/A)^2/(1-x/A - 5^2*(x/A)^2/(1-x/A -...)))))), a recursive continued fraction. [From Paul D. Hanna, Jan 05 2012]

%e G.f.: A(x) = 1 + x + x^2 + 3*x^4 + 38*x^6 + 947*x^8 + 37394*x^10 +...

%e where G(x) = A(x*G(x)) is the o.g.f. of A003701:

%e G(x) = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 +...

%e while the e.g.f. of A003701 is given by:

%e exp(x)/cos(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 12*x^4/4! + 36*x^5/5! +...

%o (PARI) {a(n)=local(X=x+x*O(x^n),G=sum(m=0,n,m!*polcoeff(exp(X)/cos(X),m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}

%Y Cf. A003701, A158119, A157308, A157310.

%K nonn

%O 0,5

%A _Paul D. Hanna_, Feb 07 2010

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Last modified February 23 01:54 EST 2024. Contains 370265 sequences. (Running on oeis4.)