OFFSET
0,3
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 205*x^5 + 3833*x^6 +...
A(4*x)^(1/2) = 1 + 2*x + 14*x^2 + 132*x^3 + 2582*x^4 + 97948*x^5 +...
A(9*x)^(1/3) = 1 + 3*x + 45*x^2 + 936*x^3 + 42255*x^4 +...
A(16*x)^(1/4) = 1 + 4*x + 104*x^2 + 3808*x^3 + 309856*x^4 +...
A(25*x)^(1/5) = 1 + 5*x + 200*x^2 + 11375*x^3 + 1458750*x^4 +...
A(36*x)^(1/6) = 1 + 6*x + 342*x^2 + 27900*x^3 + 5182758*x^4 +...
A(49*x)^(1/7) = 1 + 7*x + 539*x^2 + 59682*x^3 + 15155112*x^4 +...
A(64*x)^(1/8) = 1 + 8*x + 800*x^2 + 115456*x^3 + 38417920*x^4 +...
A(81*x)^(1/9) = 1 + 9*x + 1134*x^2 + 206793*x^3 + 87311601*x^4 +...
A(100*x)^(1/10) = 1 + 10*x + 1550*x^2 + 348500*x^3 + 182033750*x^4 +...
A(121*x)^(1/11) = 1 + 11*x + 2057*x^2 + 559020*x^3 + 353916893*x^4 +...
A(144*x)^(1/12) = 1 + 12*x + 2664*x^2 + 860832*x^3 + 649514592*x^4 +...
...
Initial terms equal the antidiagonal sums of above coefficients:
a(1) = 1 ;
a(2) = 1 + 1 = 2 ;
a(3) = 1 + 2 + 2 = 5 ;
a(4) = 1 + 3 + 14 + 5 = 23 ;
a(5) = 1 + 4 + 45 + 132 + 23 = 205 ;
a(6) = 1 + 5 + 104 + 936 + 2582 + 205 = 3833 ;
a(7) = 1 + 6 + 200 + 3808 + 42255 + 97948 + 3833 = 148051 ; ...
PROG
(PARI) {a(n)=local(A=1+x); for(n=2, n, A=1 + sum(k=1, n, x^k*subst(A, x, k^2*x+x*O(x^n))^(1/k))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 02 2009
STATUS
approved