OFFSET
0,3
COMMENTS
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) (2*n) * [x^n] A(x)^(2*n) = (2*n-1) * [x^n] A(x)^(2*n+1) for n > 1.
(2) [x^n] A(x)^(2*n) = A396846(n) for n >= 1, where A396846 forms the logarithmic derivative of A397245.
(3) [x^n] A(x)^(n+1) = (n+1) * A397245(n) for n >= 0, where Sum_{n>=0} A397245(n)*x^n = exp(x + Sum_{n>=2} (4*n^2-1) * A397245(n)*x^n / (4*n^2) ).
(4) A(x) = F(x*A(x)^2) where F(x) = A(x/F(x)^2) is the g.f. of A397245.
EXAMPLE
G.f.: A(x) = 1 + x + 6*x^2 + 235*x^3 + 17010*x^4 + 1912023*x^5 + 302087720*x^6 + 63489224916*x^7 + 17121352485210*x^8 + ...
The table of coefficients of x^k in A(x)^n begins
n = 1: [1, 1, 6, 235, 17010, 1912023, ...];
n = 2: [1, 2, 13, 482, 34526, 3860886, ...];
n = 3: [1, 3, 21, 742, 52566, 5847402, ...];
n = 4: [1, 4, (30), 1016, 71149, 7872408, ...];
n = 5: [1, 5, (40), 1305, 90295, 9936766, ...];
n = 6: [1, 6, 51, (1610), 110025, 12041364, ...];
n = 7: [1, 7, 63, (1932), 130361, 14187117, ...];
n = 8: [1, 8, 76, 2272, (151326), 16374968, ...];
n = 9: [1, 9, 90, 2631, (172944), 18605889, ...];
n =10: [1, 10, 105, 3010, 195240, (20880882), ...];
n =11: [1, 11, 121, 3410, 218240, (23200980), ...];
...
from which we may illustrate the definition
(2*n)*[x^n] A(x)^(2*n) = (2*n-1)*[x^n] A(x)^(2*n+1)
by
n = 2: 4*[x^2] A(x)^4 = 4*30 = 3*40 = 3*[x^2] A(x)^5;
n = 3: 6*[x^3] A(x)^6 = 6*1610 = 5*1932 = 5*[x^3] A(x)^7;
n = 4: 8*[x^4] A(x)^8 = 8*151326 = 7*172944 = 7*[x^4] A(x)^9;
n = 5: 10*[x^5] A(x)^10 = 10*20880882 = 9*23200980 = 9*[x^5] A(x)^11;
etc.
Also, we obtain A397245 from a semi-diagonal
A397245 = [1/1, 3/3, 40/5, 1932/7, 172944/9, 23200980/11, ...]
= [1, 1, 8, 276, 19216, 2109180, 327968280, ...].
Further, we obtain A396846 from half a semi-diagonal
A396846 = (1/2)*[2, 30, 1610, 151326, 20880882, ...]
= [1, 15, 805, 75663, 10440441, ...]
which forms the logarithmic derivative of A397245.
PROG
(PARI) \\ returns vector of terms a(k) for k = 0..n
{a(n) = my(A=[1, 1], m); for(i=1, n-1, A=concat(A, 0); m=#A-1;
A[#A] = (2*m-1)*polcoef(Ser(A)^(2*m+1), m) - 2*m*polcoef(Ser(A)^(2*m), m) ); GF=Ser(A); A}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 24 2026
STATUS
approved