A352818 G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(2*n) + (-1)^n*A(x))^n.

1, 2, 3, 5, 10, 21, 45, 100, 225, 515, 1190, 2780, 6550, 15555, 37187, 89428, 216163, 524890, 1279727, 3131526, 7688383, 18933352, 46754406, 115750501, 287238398, 714340089, 1780091533, 4444198347, 11114847215, 27843451884, 69856594565, 175515654498

Offset

1,2

Links

Table of n, a(n) for n=1..32.

Formula

G.f. A(x) satisfies:

(1) 1 - x = Sum_{n>=0} ( x^(2*n) + (-1)^n*A(x) )^n.

(2) 1 - x = Sum_{n>=0} x^(2*n^2) / (1 + (-1)^n*x^(2*n)*A(x))^(n+1).

Example

G.f.: A(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 10*x^5 + 21*x^6 + 45*x^7 + 100*x^8 + 225*x^9 + 515*x^10 + 1190*x^11 + 2780*x^12 + ...
where
1 - x = 1 + (x^2 - A(x)) + (x^4 + A(x))^2 + (x^6 - A(x))^3 + (x^8 + A(x))^4 + (x^10 - A(x))^5 + (x^12 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^2/(1 - x^2*A(x))^2 + x^8/(1 + x^4*A(x))^3 + x^18/(1 - x^6*A(x))^4 + x^32/(1 + x^8*A(x))^5 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (x^(2*m) + (-1)^m*x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A\2), x^(2*m^2)/(1 + (-x^2)^m*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A317997, A352817, A352819, A352820, A352821.

Sequence in context: A218532 A352879 A022861 * A346970 A001646 A103595

Adjacent sequences: A352815 A352816 A352817 * A352819 A352820 A352821

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved