A143926 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*A(x)^2*A(-x)^2.
1, 1, 1, 1, 2, 3, 7, 11, 28, 46, 123, 207, 572, 979, 2769, 4797, 13806, 24138, 70414, 123998, 365636, 647615, 1926505, 3428493, 10273870, 18356714, 55349155, 99229015, 300783420, 540807165, 1646828655, 2968468275, 9075674700
Offset: 0
Keywords
Examples
G.f. A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 7*x^6 + 11*x^7 +... A(x)*A(-x) = 1 + x^2 + 3*x^4 + 11*x^6 + 46*x^8 + 207*x^10 + 979*x^12 +... A(x)^2*A(-x)^2 = 1 + 2*x^2 + 7*x^4 + 28*x^6 + 123*x^8 + 572*x^10 +... A(x)^4*A(-x)^4 = 1 + 4*x^2 + 18*x^4 + 84*x^6 + 407*x^8 + 2028*x^10 +... from this we see that if B(x^2) = A(x)*A(-x) then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4 and A(x) = 1 + x*B(x^2) + x^2*B(x^2)^2.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..100
Programs
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Mathematica
a[n_] := Module[{A = 1 + x, B}, For[i = 0, i <= n, i++, B = A*(A /. x -> -x); A = 1 + x*B + x^2*B^2 + O[x]^(n+1) // Normal]; SeriesCoefficient[A, {x, 0, n}]]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 22 2016, adapted from PARI *) -
PARI
{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,B=A*subst(A,x,-x);A=1+x*B+x^2*B^2);polcoeff(A,n)}
Formula
Define B(x) by B(x^2) = A(x)*A(-x); then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4 is the g.f. of A006605.
Recurrence: 3*n*(3*n+1)*(3*n+2)*(507*n^4 - 3575*n^3 + 8895*n^2 - 8953*n + 3054)*a(n) = - 12*(4017*n^5 - 20319*n^4 + 31895*n^3 - 17595*n^2 + 2338*n + 384)*a(n-1) + 4*(n-2)*(17745*n^6 - 125125*n^5 + 331891*n^4 - 396335*n^3 + 173912*n^2 + 17532*n - 13140)*a(n-2) - 144*(n-3)*(n-2)*(312*n^3 - 988*n^2 + 407*n + 29)*a(n-3) + 144*(n-4)*(n-3)*(n-2)*(507*n^4 - 1547*n^3 + 1212*n^2 + 140*n - 72)*a(n-4). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = sqrt(70/27+(26*sqrt(13))/27) = 2.4626418602647616787... is the root of the equation -144 - 140*d^2 + 27*d^4 = 0 and c = 2*sqrt((5+1/sqrt(13))/3)/3 = 0.88421131194123... if n is even, and c = sqrt(1+11/sqrt(13))/3 = 0.670890873659690... if n is odd. - Vaclav Kotesovec, Dec 21 2013
Comments