A196195 G.f. satisfies: A(x) = Sum_{n>=0} x^n*(A(x)^n + A(-x)^n)/2.
1, 1, 1, 2, 4, 11, 24, 70, 163, 496, 1199, 3746, 9291, 29572, 74712, 241078, 617612, 2013979, 5215990, 17151362, 44809646, 148345585, 390352720, 1299558642, 3440133086, 11507046875, 30615708092, 102821083832, 274759034372, 925976586616, 2483758221924
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 24*x^6 + 70*x^7 + ... where A(x) = 1 + x*(A(x) + A(-x))/2 + x^2*(A(x)^2 + A(-x)^2)/2 + x^3*(A(x)^3 + A(-x)^3)/2 + x^4*(A(x)^4 + A(-x)^4)/2 + x^5*(A(x)^5 + A(-x)^5)/2 + ... Related expansions begin: (A(x) + A(-x))/2 = 1 + x^2 + 4*x^4 + 24*x^6 + 163*x^8 + 1199*x^10 + ... (A(x)^2 + A(-x)^2)/2 = 1 + 3*x^2 + 13*x^4 + 82*x^6 + 574*x^8 + ... (A(x)^3 + A(-x)^3)/2 = 1 + 6*x^2 + 30*x^4 + 199*x^6 + 1443*x^8 + ... (A(x)^4 + A(-x)^4)/2 = 1 + 10*x^2 + 59*x^4 + 414*x^6 + 3121*x^8 + ... (A(x)^5 + A(-x)^5)/2 = 1 + 15*x^2 + 105*x^4 + 785*x^6 + 6170*x^8 + ... (A(x)^6 + A(-x)^6)/2 = 1 + 21*x^2 + 174*x^4 + 1395*x^6 + 11460*x^8 + ... ... G.f. A = A(x) is a solution to the 5th degree polynomial equation: 0 = (x - 3) + (7 - 4*x)*A - (4 - 5*x^2)*A^2 + 4*(1 - 3*x)*x*A^3 + 4*(1 + x)*x^2*A^4 - (4 - x)*x^3*A^5.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..100
Crossrefs
Cf. A196196.
Programs
-
PARI
{a(n)=local(A=1+x); for(k=1, n, A=(1/(1-x*A+x*O(x^n))+1/(1-x*subst(A,x,-x+x*O(x^n))))/2); polcoeff(A, n)} for(n=0,30, print1(a(n),", ")) -
PARI
{a(n)=local(A=1+x+x*O(x^n)); for(k=1, n, A=1+sum(j=1, n, x^j*(A^j+subst(A, x,-x)^j)/2)); polcoeff(A, n)} for(n=0,30, print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) A(x) = ( 1/(1 - x*A(x)) + 1/(1 - x*A(-x)) )/2.
(2) 0 = (x - 3) + (-4*x + 7)*A(x) + (5*x^2 - 4)*A(x)^2 + (-12*x^2 + 4*x)*A(x)^3 + (4*x^3 + 4*x^2)*A(x)^4 + (x^4 - 4*x^3)*A(x)^5.
(3) 0 = A^5*x^4 + 4*(1 - A)*A^4*x^3 + (5 - 12*A + 4*A^2)*A^2*x^2 + (1 - 4*A + 4*A^3)*x - (3 - 7*A + 4*A^2), where A = A(x). - Paul D. Hanna, Jan 08 2023
(4) x*A(-x) = (2 - (2+x)*A(x) + 2*x*A(x)^2) / (1 - 2*A(x) + 2*x*A(x)^2). - Paul D. Hanna, Jan 08 2023
Comments