A295762 G.f. A(x) satisfies: A(x - 2*A(x^2)) = x + A(x^2).
1, 3, 12, 69, 444, 3060, 22104, 165195, 1266636, 9908196, 78760920, 634379124, 5166150000, 42465716328, 351876854448, 2936058188877, 24648274487556, 208040487845076, 1764376309044792, 15027939263874132, 128495423551583664, 1102547377746843624, 9490542912076091184, 81931260285359287812, 709199467337528862768, 6153967855892699398368, 53521531522907694320928, 466461452477641527148344
Offset: 1
Keywords
Examples
G.f.: A(x) = x + 3*x^2 + 12*x^3 + 69*x^4 + 444*x^5 + 3060*x^6 + 22104*x^7 + 165195*x^8 + 1266636*x^9 + 9908196*x^10 + 78760920*x^11 + 634379124*x^12 +... such that A(x - 2*A(x^2)) = x + A(x^2). RELATED SERIES. A(x - 2*A(x^2)) = x + x^2 + 3*x^4 + 12*x^6 + 69*x^8 + 444*x^10 + 3060*x^12 + 22104*x^14 + 165195*x^16 + 1266636*x^18 + 9908196*x^20 +... which equals x + A(x^2). Series_Reversion( x - 2*A(x^2) ) = x + 2*x^2 + 8*x^3 + 46*x^4 + 296*x^5 + 2040*x^6 + 14736*x^7 + 110130*x^8 + 844424*x^9 + 6605464*x^10 + 52507280*x^11 + 422919416*x^12 +... which equals (2*A(x) + x)/3. A( (x + 2*A(x))^2/9 ) = x^2 + 4*x^3 + 23*x^4 + 148*x^5 + 1020*x^6 + 7368*x^7 + 55065*x^8 + 422212*x^9 + 3302732*x^10 + 26253640*x^11 + 211459708*x^12 +... which equals (A(x) - x)/3. Odd terms seem to occur only at positions 2^n, n>=0, beginning: [1, 3, 69, 165195, 2936058188877, 2740954751925406954539018771, 6899036855844990995854505818787102393537422152828959745477, ...].
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..1030
Programs
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Mathematica
nmax = 28; sol = {a[1] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 1, n}] /. sol; eq = CoefficientList[A[x - 2 A[x^2]] - (x + A[x^2]) + O[x]^(n+1) // Normal, x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}]; sol /. Rule -> Set; a /@ Range[nmax] (* Jean-François Alcover, Nov 03 2019 *) -
PARI
{a(n) = my(A=x); for(i=1,n, A = -x/2 + 3/2*serreverse(x - 2*subst(A,x,x^2) +x^2*O(x^n)) ); polcoeff(A,n)} for(n=1,30,print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) A(x) = x + 3 * A( (x + 2*A(x))^2/9 ).
(2) A(x) = -x/2 + 3/2*Series_Reversion( x - 2*A(x^2) ).
(3) x = A( -2*x + 3 * Series_Reversion( x + A(x^2) ) ).
(4) A(x - 2*A(x^2)) = x + A(x^2).
a(n) ~ c * d^n / n^(3/2), where d = 9.2093789571628170578048631587867514135871993630966974111785256401203919511... and c = 0.0682994378249920600052979267397719638823914060552824374... - Vaclav Kotesovec, Dec 04 2017
Comments