|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + 6*x^2 + 150*x^3 + 15684*x^4 + 6626832*x^5 +...
ILLUSTRATION OF INITIAL TERMS.
If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
A^0: [1],0, 0, 0, 0, 0, 0, 0, ...;
A^1: [1, 1], 6, 150, 15684, 6626832, 11412679110, 80341130055678, ...;
A^2: [1, 2, 13], 312, 31704, 13286832, 22838822592, 160705169696760, ...;
A^3: [1, 3, 21, 487], 48078, 19980558, 34278483114, 241092139452066, ...;
A^4: [1, 4, 30, 676, 64825], 26708592, 45731714160, 321502059924816, ...;
A^5: [1, 5, 40, 880, 81965, 33471541], 57198570060, 401934951793740, ...;
A^6: [1, 6, 51, 1100, 99519, 40270038, 68679106021], 482390835814224, ...;
A^7: [1, 7, 63, 1337, 117509, 47104743, 80173378159, 562869732819493], ...; ...
then the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals 2^(n^2):
2^0 = 1 = 1;
2^1 = 1 + 1 = 2;
2^4 = 1 + 2 + 13 = 16;
2^9 = 1 + 3 + 21 + 487 = 512;
2^16 = 1 + 4 + 30 + 676 + 64825 = 65536;
2^25 = 1 + 5 + 40 + 880 + 81965 + 33471541 = 33554432;
2^36 = 1 + 6 + 51 + 1100 + 99519 + 40270038 + 68679106021 = 68719476736; ...
|
|
|
PROG
|
(PARI) /* By Definition (slow): */
{a(n)=if(n==0, 1, (2^(n^2) - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)}
for(n=0, 20, print1(a(n)*1!, ", "))
(PARI) /* Faster, using series reversion: */
{a(n)=local(B=sum(k=0, n+1, 2^(k^2)*x^k)+x^3*O(x^n), G=1+x*O(x^n));
for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); polcoeff(x/serreverse(x*G), n)}
for(n=0, 20, print1(a(n), ", "))
|
