OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..80
EXAMPLE
G.f.: A(x) = 1 + 2*x + 10*x^2 + 316*x^3 + 49286*x^4 + 29159004*x^5 + 64306390660*x^6 + 545236870010872*x^7 + 18158564638452610374*x^8 + 2398983772627848027521708*x^9 + 1262702849939184484481521481260*x^10 +...
such that the coefficient of x^n in A(x)^(2^n) equals 2^(n*(n+1)) for n>=0.
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^n in A(x)^(2^n) begins:
n=0: [1, 2, 10, 316, 49286, 29159004, 64306390660, ...];
n=1: [1, 4, 24, 672, 99936, 58521472, 128730502912, ...];
n=2: [1, 8, 64, 1536, 205824, 117874688, 257934426112, ...];
n=3: [1, 16, 192, 4096, 440320, 239239168, 517783552000, ...];
n=4: [1, 32, 640, 14336, 1048576, 494141440, 1043408617472, ...];
n=5: [1, 64, 2304, 69632, 3424256, 1073741824, 2119989985280, ...];
n=6: [1, 128, 8704, 434176, 21069824, 2906652672, 4398046511104, ...];
n=7: [1, 256, 33792, 3096576, 229048320, 18765316096, 10095488401408, 72057594037927936, ...]; ...
in which the main diagonal equals 2^(n^2+n):
[1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, ..., 4^(n*(n+1)/2), ...].
RELATED SERIES.
log(A(x)) = 2*x + 16*x^2/2 + 896*x^3/3 + 194560*x^4/4 + 145293312*x^5/5 + 385486422016*x^6/6 + 3815756479332352*x^7/7 + 145259790155527487488*x^8/8 + 21590527069867423236620288*x^9/9 + 12626980518625294860075743051776*x^10/10 +...
MATHEMATICA
terms = 15; A[x_] = Sum[a[n]*x^n, {n, 0, terms-1}];
c[n_] := Coefficient[A[x]^(2^n), x, n] == 2^(n^2+n) // Solve // First;
Do[A[x_] = (A[x] /. c[n]) + O[x]^terms, {n, 0, terms-1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 14 2018 *)
PROG
(PARI) {a(n) = my(A=[1]); for(m=1, n, A = concat(A, 0); V = Vec( Ser(A)^(2^m) ); A[m+1] = 2^(m^2) - V[m+1]/2^m; ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 05 2018
STATUS
approved