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A337720 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*(1+x)^n)^n * x^n / A(x)^n. 1
1, 1, 1, 1, 2, 4, 8, 22, 63, 190, 618, 2119, 7617, 28571, 111271, 448704, 1869072, 8024648, 35445048, 160820201, 748525951, 3569856592, 17426990777, 86999613910, 443782312584, 2311252408353, 12281279591605, 66538929813869, 367350691232741, 2065442073525419 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (1 + x*(1+x)^n)^n * x^n / A(x)^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * x^(2*n) / ( A(x) - x*(1+x)^n )^(n+1).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + 22*x^7 + 63*x^8 + 190*x^9 + 618*x^10 + 2119*x^11 + 7617*x^12 + ...
where
1 = 1/A(x) + (1 + x*(1+x))*x/A(x)^2 + (1 + x*(1+x)^2)^2*x^2/A(x)^3 + (1 + x*(1+x)^3)^3*x^3/A(x)^4 + (1 + x*(1+x)^4)^4*x^4/A(x)^5 + ... + (1 + x*(1+x)^n)^n*x^n/A(x)^(n+1) + ...
also
1 = 1/(A(x) - x) + (1+x)*x^2/(A(x) - x*(1+x))^2 + (1+x)^4*x^4/(A(x) - x*(1+x)^2)^3 + (1+x)^9*x^6/(A(x) - x*(1+x)^3)^4 + (1+x)^16*x^8/(A(x) - x*(1+x)^4)^5 + (1+x)^25*x^10/(A(x) - x*(1+x)^5)^6 + ... + (1+x)^(n^2)*x^(2*n)/(A(x) - x*(1+x)^n)^(n+1) + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (1 + x*(1+x)^m +x*O(x^#A))^m * x^m / Ser(A)^(m+1) ), #A-1)); A[#A]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (1+x +x*O(x^#A))^(m^2) * x^(2*m) / ( Ser(A) - x*(1+x)^m )^(m+1) ), #A-1)); A[#A]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A337721.
Sequence in context: A027385 A158324 A002075 * A293912 A122623 A124014
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 16 2020
STATUS
approved

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Last modified April 17 13:58 EDT 2024. Contains 371764 sequences. (Running on oeis4.)