|
|
A299434
|
|
G.f. A(x) satisfies: 1 = Sum_{n>=0} binomial((n+1)^2,n)/(n+1)^2 * x^n / A(x)^((n+1)^2).
|
|
2
|
|
|
1, 1, 1, 6, 77, 1451, 35730, 1082481, 38913817, 1619979291, 76724619427, 4077896446598, 240566693095072, 15609120639706252, 1105414601508493001, 84881459931003622118, 7026832554316541379141, 624014794413319426058889, 59184228450018585954486975, 5971678912361406721742217080, 638782082648832471805820934833, 72213308562202419209594988387550, 8603323896642095980014195130664418
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
Compare to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.
|
|
LINKS
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 77*x^4 + 1451*x^5 + 35730*x^6 + 1082481*x^7 + 38913817*x^8 + 1619979291*x^9 + 76724619427*x^10 +...
such that
1 = 1/A(x) + C(4,1)/4*x/A(x)^4 + C(9,2)/9*x^2/A(x)^9 + C(16,3)/16*x^3/A(x)^16 + C(25,4)/25*x^4/A(x)^25 + C(36,5)/36*x^5/A(x)^36 + C(49,6)/49*x^6/A(x)^49 + ...
more explicitly,
1 = 1/A(x) + x/A(x)^4 + 4*x^2/A(x)^9 + 35*x^3/A(x)^16 + 506*x^4/A(x)^25 + 10472*x^5/A(x)^36 + 285384*x^6/A(x)^49 + ... + A143669(n)*x^n/A(x)^((n+1)^2) + ...
|
|
PROG
|
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = Vec(sum(n=0, #A, binomial((n+1)^2, n)/(n+1)^2 * x^n/Ser(A)^((n+1)^2-1) ))); G=Ser(A); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|