A359463 Coefficient a(n) of x^n in power series A(x), n >= 0, such that A(x) = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^(n-1))^n.

1, 1, 2, 6, 20, 69, 245, 896, 3362, 12869, 50024, 196896, 783205, 3143713, 12717532, 51798089, 212233756, 874193355, 3617797596, 15035379576, 62724649455, 262579756558, 1102680011825, 4643936681122, 19609621413193, 83005706694022, 352145760387515, 1497067760933244

Offset: 0

Author

Paul D. Hanna, Jan 17 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 245*x^6 + 896*x^7 + 3362*x^8 + 12869*x^9 + 50024*x^10 + 196896*x^11 + 783205*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.22210374835192555734961892166866769267669905135315...
A(1/5) = 1.45174689360673617561694352881716190508117725206270...
A(1/6) = 1.28852385900727494844427701605174847197781970881818...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A290003.

Programs

PARI

{a(n) = my(A=1);
A = (-1/x)*serreverse(-x/sum(m=-n-1,n+1, (x - x^m +x*O(x^(n+1)))^m )); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

PARI

{a(n) = my(A=1); for(i=1,n,
A = sum(m=-n,n, (-x*A)^m * (1 - (-x*A)^(m-1) +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

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

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

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

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

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

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

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

(8) 1/x = Sum_{n=-oo..+oo} (x*A(x))^(2*n) * (1 - (-x*A(x))^n)^n.

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

(10) 0 = Sum_{n=-oo..+oo} (x*A(x))^(2*n) * (1 - (-x*A(x))^n)^(n+1).

(11) A(-x/G(x)) = G(x) where G(x) = Sum_{n=-oo..+oo} (x - x^n)^n is the g.f. of A290003.

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

a(n) ~ c * d^n / n^(3/2), where d = 4.4911010651615255101195452998052055698... and c = 0.53507007927413038001531299966030791... - Vaclav Kotesovec, Mar 14 2023