A374570 Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*C(x) ), where C(x) = x + C(x)^2 is the Catalan function (A000108).
1, 1, 3, 8, 27, 90, 320, 1152, 4257, 15934, 60486, 231894, 897242, 3497638, 13725678, 54174286, 214923493, 856560918, 3427838222, 13768875142, 55494305328, 224359469870, 909656736876, 3697874061870, 15068978724200, 61545704828266, 251899370771284, 1033027441769384
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 90*x^6 + 320*x^7 + 1152*x^8 + 4257*x^9 + 15934*x^10 + 60486*x^11 + 231894*x^12 + ... RELATED SERIES. A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 79*x^6 + 282*x^7 + 1046*x^8 + 3916*x^9 + 14907*x^10 + 57274*x^11 + 222194*x^12 + ... where A(x)^2 = A( A(x)*C(x) ). A(x)*C(x) = x^2 + 2*x^3 + 6*x^4 + 18*x^5 + 60*x^6 + 204*x^7 + 720*x^8 + 2586*x^9 + 9468*x^10 + 35124*x^11 + 131898*x^12 + ... C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n)*x^n + ,,, where C(x) = (1 - sqrt(1 - 4*x))/2 is the Catalan function. Let B(x) satisfy A(x*B(x)) = x, then B(x) = 1 - x - x^2 + 2*x^3 - x^4 + x^5 + 2*x^6 - 6*x^7 - x^8 + 5*x^9 + x^10 + 2*x^12 - 8*x^13 - 6*x^14 + 22*x^15 - x^16 + ... + A374571(n)*x^n + ... where C(x*B(x)) = x*B(x^2) and C(x) = x + C(x)^2. Also notice that A(x-x^2) is the odd function starting as A(x-x^2) = x + x^3 + 4*x^5 + 18*x^7 + 96*x^9 + 546*x^11 + 3274*x^13 + 20326*x^15 + 129622*x^17+ 843854*x^19 + ... satisfying A(x-x^2)^2 = A( x*A(x-x^2) ). SPECIFIC VALUES. G.f. A(x) diverges at x = 1/4; what is the radius of convergence? A(2/9) = 0.410501753930478190014767562028185186269192589705662553072... where A(2/9)^2 = A( (1/3) * A(2/9) ). A(1/5) = 0.307823207567908585715446000098072863270477544252476707540... where A(1/5)^2 = A( A(1/5) * (1 - sqrt(1/5))/2 ). A(1/6) = 0.222895676073964945442191376315546188067098171316653068516... where A(1/6)^2 = A( A(1/6) * (1 - sqrt(1/3))/2 ). A(1/8) = 0.149886223456626114071674919752683973970671151550604884301... where A(1/8)^2 = A( A(1/8) * (1 - sqrt(1/2))/2 ). A(1/10) = 0.11421035457722945538609562679806658343632346343476019471... where A(1/10)^2 = A( A(1/10) * (1 - sqrt(3/5))/2 ). A(1/12) = 0.09255115114959352826965125804331807348315032543228258146... where A(1/12)^2 = A( A(1/12) * (1 - sqrt(2/3))/2 ).
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..1030
Programs
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PARI
{a(n) = my(A=[1], Ax, C = serreverse(x-x^2 + x^2*O(x^n))); for(i=1, n, A=concat(A, 0); Ax=x*Ser(A); A[#A] = -polcoeff( Ax^2 - subst(Ax, x, Ax*C ), #A+1) ); A[n]} for(n=1, 30, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n)*x^n, where B(x) = (1/x)*Series_Reversion(A(x)) is the g.f. of A374571 and C(x) = x + C(x)^2 is the g.f. of A000108, satisfies the following formulas.
(1) A(x)^2 = A( A(x)*C(x) ).
(2) x^2 = A( x*C(x*B(x)) ).
(3) A(x) = x / B(A(x)).
(4) A(x) = C(x) / B(A(x)^2).
(5) C(x*B(x)) = x*B(x^2).
(6) A(x-x^2)^2 = A( x*A(x-x^2) ).
Comments