A363142 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).

1, 1, 3, 7, 17, 42, 107, 275, 715, 1884, 5009, 13421, 36224, 98382, 268657, 737244, 2032035, 5622938, 15615186, 43505382, 121570407, 340639265, 956861955, 2694064938, 7601455079, 21490621769, 60870280259, 172707869088, 490818655346, 1396973741672, 3981748142925

Offset: 0

Author

Paul D. Hanna, May 17 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 42*x^5 + 107*x^6 + 275*x^7 + 715*x^8 + 1884*x^9 + 5009*x^10 + 13421*x^11 + 36224*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A359711, A363143, A363144, A363182.

Cf. A359670.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (Ser(A) + x^(2*m-1))^(m+1) ),#A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.

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

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

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

(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (A(x) + x^(2*n-1))^(n-1).

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

a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) for n >= 0. - Paul D. Hanna, May 18 2023