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A306763
G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n^2) * x^n = Sum_{n>=0} binomial(n^2,n) * x^n.
0
1, 5, 63, 1372, 41814, 1605215, 73586824, 3906566501, 235444126392, 15881634865780, 1185873283860557, 97147220190772317, 8665813010430379775, 836342349269443514470, 86843462603384158258103, 9655074380695222955712860, 1144404915485406530977640253, 144066096386630152751096708253, 19197014710932516253131393942286, 2699479675453423906131984772100102
OFFSET
0,2
EXAMPLE
G.f.: A(x) = 1 + 5*x + 63*x^2 + 1372*x^3 + 41814*x^4 + 1605215*x^5 + 73586824*x^6 + 3906566501*x^7 + 235444126392*x^8 + 15881634865780*x^9 + ...
such that the following series are equal:
B(x) = 1 + A(x)*x + A(x)^4*x^2 + A(x)^9*x^3 + A(x)^16*x^4 + A(x)^25*x^5 + A(x)^36*x^6 + A(x)^49*x^7 + A(x)^64*x^8 + ...
B(x) = 1 + x + 6*x^2 + 84*x^3 + 1820*x^4 + 53130*x^5 + 1947792*x^6 + 85900584*x^7 + 4426165368*x^8 + 260887834350*x^9 + ... + binomial(n^2,n) * x^n + ...
MATHEMATICA
a[n_] := Module[{A = {1}}, For[i = 1, i <= n, i++, AppendTo[A, 0]; A[[-1]] = -Coefficient[Sum[x^m*(A.x^Range[0, Length[A]-1])^(m^2) - x^m* Binomial[m^2, m], {m, 0, Length[A]}], x, Length[A]]]; A[[n+1]] ];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, May 07 2019, from PARI *)
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = -polcoeff( sum(m=0, #A, x^m*Ser(A)^(m^2) - x^m*binomial(m^2, m) ), #A) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Cf. A014062 (binomial(n^2,n))).
Sequence in context: A355411 A334907 A218102 * A275763 A004193 A193326
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 04 2019
STATUS
approved