OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..600
EXAMPLE
G.f.: 1 = ... ((((((((1/(1-x) - 1*x)^2 - 2*x^2)^2 - 4*x^3)^2 - 12*x^4)^2 - 32*x^5)^2 - 112*x^6)^2 - 384*x^7)^2 - 1824*x^8)^2 -...- a(n)*x^n)^2 -...
ILLUSTRATION OF GENERATING METHOD.
Start with G1 = 1/(1-x), and proceed as follows:
G2 = (G1 - 1*x)^2 = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 +...
G3 = (G2 - 2*x^2)^2 = 1 + 4*x^3 + 6*x^4 + 8*x^5 + 14*x^6 + 24*x^7 + 39*x^8 +...
G4 = (G3 - 4*x^3)^2 = 1 + 12*x^4 + 16*x^5 + 28*x^6 + 48*x^7 + 114*x^8 + 216*x^9 +...
G5 = (G4 - 12*x^4)^2 = 1 + 32*x^5 + 56*x^6 + 96*x^7 + 228*x^8 + 432*x^9 +...
G6 = (G5 - 32*x^5)^2 = 1 + 112*x^6 + 192*x^7 + 456*x^8 + 864*x^9 + 2144*x^10 +...
G7 = (G6 - 112*x^6)^2 = 1 + 384*x^7 + 912*x^8 + 1728*x^9 + 4288*x^10 + 9664*x^11 +...
G8 = (G7 - 384*x^7)^2 = 1 + 1824*x^8 + 3456*x^9 + 8576*x^10 + 19328*x^11 +...
G9 = (G8 - 1824*x^8)^2 = 1 + 6912*x^9 + 17152*x^10 + 38656*x^11 + 106560*x^12 +...
G10 = (G9 - 6912*x^9)^2 = 1 + 34304*x^10 + 77312*x^11 + 213120*x^12 + 532480*x^13 +...
G11 = (G10 - 34304*x^10)^2 = 1 + 154624*x^11 + 426240*x^12 + 1064960*x^13 +...
...
G_{n+1} = (G_{n} - a(n)*x^n)^2 = 1 + a(n+1)*x^(n+1) + a(n+2)*x^(n+2)/2 + a(n+3)*x^(n+3)/4 + a(n+4)*x^(n+4)/8 +...
...
PROG
(PARI) {a(n) = my(A=[1], G); for(i=1, n, A=concat(A, 0); G = 1/(1-x +x*O(x^#A)); for(m=1, #A, G = (G - A[m]*x^m)^2 ); A[#A] = polcoeff(G, #A)/2 ); A[n]}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Informal quick print of initial N terms: */
{N=100; A=[1]; G = 1/(1-x +x^2*O(x^N)); for(m=1, N, A=concat(A, 0); G = (G - A[m]*x^m)^2; A[m+1] = polcoeff(G, m+1); print1(A[m], ", ")); print1(A[N], ", ")}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 12 2016
STATUS
approved