A359711 a(n) = coefficient of x^n in A(x) such that 1 = Sum_{n=-oo..+oo} (-x)^n * (A(x) + x^(n-1))^(n+1).

1, 3, 11, 42, 165, 671, 2795, 11877, 51286, 224413, 992924, 4434833, 19969030, 90550829, 413148619, 1895338362, 8737219074, 40452543831, 188025758635, 877055405522, 4104269624748, 19262955163275, 90652992751518, 427681283728070, 2022341915324936, 9583224591208298

Offset: 0

Author

Paul D. Hanna, Jan 17 2023

Keywords

• nonn

Comments

Row sums of triangle A359670.

Examples

			G.f.: A(x) = 1 + 3*x + 11*x^2 + 42*x^3 + 165*x^4 + 671*x^5 + 2795*x^6 + 11877*x^7 + 51286*x^8 + 224413*x^9 + 992924*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A359670, A359712, A359713, A363104, A363105.

Cf. A363142, A363143, A363144.

Programs

PARI

{a(n) = my(A=1,y=1); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A,n,x)}
for(n=0,25, print1( a(n),", "))

PARI

{a(n) = my(A=[1],y=1); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.

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

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

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

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

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

From Paul D. Hanna, May 18 2023: (Start)

(6) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x) + x^n)^n.

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

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

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^n)^(n+1).

(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^n)^n. (End)

a(n) = Sum_{k=0..n} A359670(n,k) for n >= 0.

a(n) ~ c * d^n / n^(3/2), where d = 5.008723344615566939692217... and c = 4.45330627132612826203... - Vaclav Kotesovec, Mar 14 2023