A374571 Expansion of g.f. A(x) satisfying A(x) = A(x^2) - x*A(x^2)^2.
1, -1, -1, 2, -1, 1, 2, -6, -1, 5, 1, 0, 2, -8, -6, 22, -1, -11, 5, -30, 1, 33, 0, 0, 2, -16, -8, 52, -6, -40, 22, -114, -1, 125, -11, 90, 5, -149, -30, 154, 1, -123, 33, -360, 0, 552, 0, 144, 2, -440, -16, 256, -8, -360, 52, -552, -6, 1176, -40, 576, 22, -1360, -114, 470, -1, -235, 125, -1710, -11, 3387, 90, 486, 5, -3353, -149, 1864, -30, -2152, 154, -2250, 1
Offset: 0
Keywords
Examples
G.f.: A(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 - 11*x^17 + 5*x^18 - 30*x^19 + x^20 + ... where A(x^2) = (1 - sqrt(1 - 4*x*A(x)))/(2*x). RELATED SERIES. Let B(x) = Series_Reversion(x*A(x)), then B(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 + ... + A374570(n)*x^n + ... where B(x)^2 = B( B(x)*C(x) ), and C(x) begins: 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. SPECIFIC VALUES. A(t) = 4/5 at t = 0.1786763406278486221896028296025274247659944115... A(t) = 3/4 at t = 0.2209727374872302749773868295900473238254186343... A(t) = 2/3 at t = 0.2927920532546611624693565662579476873870699464... A(t) = 3/5 at t = 0.3532836501852252091389612952989266014287213872... A(t) = 1/2 at t = 0.4540878993396162878365437853450173746622109652... A(t) = 2/5 at t = 0.5753264646036491718800481741299163550606457682... A(t) = 1/3 at t = 0.6711059159867924708010090309770441047524321152... A(t) = 1/4 at t = 0.8063263233032142016966341297674341884930955548... A(t) = 1/5 at t = 0.8884702348196434968520432792716046325517863531... A(1/2) = 0.4596569887547343191321148479065626411948116168891503813... where A(1/4) = (1 - sqrt(1 - 2*A(1/2))). A(1/3) = 0.6215166290026409046430206750366100166629591510407086872... where A(1/9) = (3/2) * (1 - sqrt(1 - (4/3)*A(1/3))). A(1/4) = 0.7159471484203487850228006105062270686816491955635126263... where A(1/16) = 2 * (1 - sqrt(1 - A(1/4))). A(1/5) = 0.7747713037551020088783260174094983351988173792698848600... where A(1/25) = (5/2) * (1 - sqrt(1 - (4/5)*A(1/5))). A(1/6) = 0.8141931617547219509824463958597943246122338043286847588... where A(1/36) = 3 * (1 - sqrt(1 - (2/3)*A(1/6))). A(1/8) = 0.8630723739180924020163457579861333293488991044015651008... where A(1/64) = 4 * (1 - sqrt(1 - (1/2)*A(1/8))). A(1/10) = 0.891911395101161792043000371010714789952867553398091597... where A(1/100) = 5 * (1 - sqrt(1 - (2/5)*A(1/10))).
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..5000
Programs
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PARI
{a(n) = my(A = 1+x); for(i=0,#binary(n), A = subst(A,x,x^2) - x*subst(A^2,x,x^2) + x*O(x^n) ); polcoeff(A,n)} for(n=0, 80, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n)*x^n, where B(x) is the g.f. of A374570 and C(x) = x + C(x)^2 is the g.f. of A000108, satisfies the following formulas.
(1) A(x) = A(x^2) - x*A(x^2)^2.
(2) A(x^2) = (1 - sqrt(1 - 4*x*A(x))) / (2*x).
(3) A(x^2) = (1/x) * C(x*A(x)).
(4) x^2 = B( x * C(x*A(x)) ).
(5) A(B(x)) = x / B(x).
(6) A(B(x)^2) = C(x) / B(x).
(7) B(x)^2 = B( B(x)*C(x) ).
Comments