A378571 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n + 1)^(2*n+1).

1, 1, 2, 5, 17, 56, 190, 670, 2452, 9139, 34512, 131960, 510376, 1992650, 7840845, 31063216, 123809267, 496117444, 1997461096, 8076398383, 32780982554, 133516673419, 545535083011, 2235451243956, 9184586572401, 37828126885778, 156152936538097, 645942256355057, 2677202474308391

Offset

1,3

Comments

Related identity: Sum_{n=-oo..+oo} x^(2*n) * (y - x^n)^(n+1) = 0 which holds formally for all y.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

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

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

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

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

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

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 17*x^5 + 56*x^6 + 190*x^7 + 670*x^8 + 2452*x^9 + 9139*x^10 + 34512*x^11 + 131960*x^12 + ...
If B(A(x)) = x then B(x) = 1/Sum_{n=-oo..+oo} x^(2*n)*(1 + x^n)^(2*n+1), where
1/B(x) = 1/x + 1 + x + x^2 + 4*x^3 + 2*x^5 + 11*x^6 + x^7 + 9*x^9 + 25*x^10 + x^11 + 9*x^12 + x^13 + 29*x^14 + 62*x^15 + x^17 + 58*x^18 + x^19 + 85*x^20 + 107*x^21 + 66*x^22 + x^23 + ...
so that 1/x = 1/A(x) + 1 + A(x) + A(x)^2 + 4*A(x)^3 + 2*A(x)^5 + 11*A(x)^6 + A(x)^7 + 9*A(x)^9 + 25*A(x)^10 + A(x)^11 + 9*A(x)^12 + A(x)^13 + 29*A(x)^14 + ...
SPECIFIC VALUES.
A(t) = 2/5 at t = 0.22786803543899859355398486947189346245516141048897...
  where t = 1/Sum_{n=-oo..+oo} 4^n * (5^n + 2^n)^(2*n+1) / 5^(n*(2*n+3)).
A(t) = 1/3 at t = 0.21657754401903945978657792097322947429225492866288...
  where t = 1/Sum_{n=-oo..+oo} (3^n + 1)^(2*n+1) / 3^(n*(2*n+3)),
  also t = -1/Sum_{n=-oo..+oo} (3^n - 1)^(2*n+1) / 3^(n*(2*n+3)).
A(t) = 1/4 at t = 0.18588193823667927186096739805587668626707432037365...
  where t = 1/Sum_{n=-oo..+oo} (4^n + 1)^(2*n+1) / 4^(n*(2*n+3)),
  also t = -1/Sum_{n=-oo..+oo} (4^n - 1)^(2*n+1) / 4^(n*(2*n+3)).
A(t) = 1/5 at t = 0.15940410834528665053792581982752377225328045286010...
  where t = 1/Sum_{n=-oo..+oo} (5^n + 1)^(2*n+1) / 5^(n*(2*n+3)),
  also t = -1/Sum_{n=-oo..+oo} (5^n - 1)^(2*n+1) / 5^(n*(2*n+3)).
A(1/5) = 0.28268682044400737132370699087845721953093955202610...
  where 5 = Sum_{n=-oo..+oo} A(1/5)^(2*n) * (A(1/5)^n + 1)^(2*n+1).
A(1/6) = 0.21269063547687973263079039900242231172049186993028...
A(1/7) = 0.17314523426815121829552118154260083141856977455281...
A(1/8) = 0.14664945265382114740665503553786883889728617002117...
A(1/10) = 0.1127556269613735651181342219674337035916106838755...
A(-1/5) = -0.1712938380186308563886088555938903545042719417719...

Prog

(PARI) N=40 \\ number of terms
{a(n) = my(R = sum(m=-N-1, N+1, x^(2*m) * (1 + x^m +x^2*O(x^N))^(2*m+1) ), A);
A = serreverse(1/R); polcoef(A, n)}
for(n=1, N, print1(a(n), ", "))

Crossrefs

Sequence in context: A159866 A042671 A180148 * A241133 A148410 A190531

Adjacent sequences: A378568 A378569 A378570 * A378572 A378573 A378574

Keyword

nonn

Author

Paul D. Hanna, Dec 29 2024

Status

approved