OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n) )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(4*n+4).
(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n,
then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).
(5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n,
then C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).
(6) Let D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n,
then D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).
a(n) ~ 2^(2*n + log(2)/8 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018
EXAMPLE
G.f.: A(x) = 1 + 4*x + 26*x^2 + 356*x^3 + 8871*x^4 + 320672*x^5 + 14811200*x^6 + 820185072*x^7 + 52546341422*x^8 + 3808527303300*x^9 + 307523461730866*x^10 + ...
such that
1 = 1 + (1/A(x) - (1-x)^4) + (1/A(x) - (1-x)^8)^2 + (1/A(x) - (1-x)^12)^3 + (1/A(x) - (1-x)^16)^4 + (1/A(x) - (1-x)^20)^5 + (1/A(x) - (1-x)^24)^6 + (1/A(x) - (1-x)^28)^7 + (1/A(x) - (1-x)^32)^8 + ...
Also,
A(x) = 1 + (1/A(x) - (1-x)^8) + (1/A(x) - (1-x)^12)^2 + (1/A(x) - (1-x)^16)^3 + (1/A(x) - (1-x)^20)^4 + (1/A(x) - (1-x)^24)^5 + (1/A(x) - (1-x)^28)^6 + (1/A(x) - (1-x)^32)^7 + (1/A(x) - (1-x)^36)^8 + ...
RELATED SERIES.
(1) The series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n begins
B(x) = 1 + x + 5*x^2 + 67*x^3 + 1669*x^4 + 60246*x^5 + 2781335*x^6 + 154062232*x^7 + 9875799121*x^8 + 716231200582*x^9 + 57865799711347*x^10 + ...
also given by B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).
(2) The series C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n begins
C(x) = 1 + 2*x + 11*x^2 + 148*x^3 + 3683*x^4 + 132888*x^5 + 6131332*x^6 + 339397944*x^7 + 21742672693*x^8 + 1575995237188*x^9 + 127268039660042*x^10 + ...
also given by C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).
(3) The series D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n begins
D(x) = 1 + 3*x + 18*x^2 + 244*x^3 + 6073*x^4 + 219238*x^5 + 10117351*x^6 + 560000464*x^7 + 35868610134*x^8 + 2599382401532*x^9 + 209871544727484*x^10 + ...
also given by D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - (1-x)^(4*m+4) )^m ) )[#A]/2 ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 12 2018
STATUS
approved