A143549 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x).
1, 1, 3, 17, 85, 598, 3473, 26668, 166429, 1340079, 8724438, 72374714, 484498327, 4102336176, 28009706440, 240729330116, 1668007246157, 14499527706129, 101618389067849, 891275643857227, 6303425058175018, 55686806813191060
Offset: 0
Keywords
Examples
G.f. A(x) = 1 + x + 3*x^2 + 17*x^3 + 85*x^4 + 598*x^5 + 3473*x^6 +... Related expansions: A(x)^4 = 1 + 4*x + 18*x^2 + 108*x^3 + 635*x^4 + 4348*x^5 + 28336*x^6 +... A(x)*A(-x) = 1 + 5*x^2 + 145*x^4 + 5971*x^6 + 287253*x^8 +... [A(x)*A(-x)]^5 = 1 + 25*x^2 + 975*x^4 + 45605*x^6 + 2355490*x^8 +...
Links
- Robert Israel, Table of n, a(n) for n = 0..600
Programs
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Maple
S:= series(RootOf(_Z^15*x^3-_Z^12*x^2+_Z^11*x^2-_Z^4+4*_Z^3-6*_Z^2+4*_Z-1),x,31): seq(coeff(S,x,i),i=0..30); # Robert Israel, Jul 10 2017
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Mathematica
nmax = 21; sol = {a[0] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x*A[x]^4*A[-x]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}]; sol /. Rule -> Set; a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *) -
PARI
{a(n)=local(A=1+x*O(x^n));for(i=0,2*n,A=1+x*A^4*subst(A^1,x,-x));polcoeff(A,n)}
Formula
G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^5.
G.f. satisfies: -x^3*A(x)^15+x^2*A(x)^12-x^2*A(x)^11+A(x)^4-4*A(x)^3+6*A(x)^2-4*A(x)+1 = 0. - Robert Israel, Jul 10 2017
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (-1)^i * a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025