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, 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

Offset

0,5

Links

Table of n, a(n) for n=0..29.

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

Adjacent sequences: A337717 A337718 A337719 * A337721 A337722 A337723

Keyword

nonn

Author

Paul D. Hanna, Sep 16 2020

Status

approved