A379196 G.f. A(x) satisfies 2 = Sum_{n=-oo..+oo} x^(4*n) * (1 - x^n)^(4*n) * A(x)^n.

1, 4, 10, 20, 36, 60, 94, 140, 204, 292, 474, 756, 1556, 2600, 5612, 9404, 21174, 39896, 95446, 204960, 486168, 1076000, 2392306, 5215772, 11042630, 24216104, 51067130, 115069352, 246871472, 563237900, 1216798802, 2743669520, 5908953208, 13085409072, 28224118088, 61889692388, 134885809852

Offset

0,2

Comments

Compare to the identity: 0 = Sum_{n=-oo..+oo} x^(4*n) * (1 - x^n)^(n+3).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..520

Formula

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

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

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

Example

G.f.: A(x) = 1 + 4*x + 10*x^2 + 20*x^3 + 36*x^4 + 60*x^5 + 94*x^6 + 140*x^7 + 204*x^8 + 292*x^9 + 474*x^10 + 756*x^11 + 1556*x^12 + ...
where 2 = Sum_{n=-oo..+oo} x^(4*n) * (1 - x^n)^(4*n) * A(x)^n.
SPECIFIC VALUES.
A(t) = 2 at t = 0.158968883431737234519930958476417749332153250593476...
  where 2 = Sum_{n=-oo..+oo} 2^n * t^(4*n) * (1 - t^n)^(4*n).
A(t) = 3 at t = 0.239523536691898630692922508800842750666883765064577...
  where 2 = Sum_{n=-oo..+oo} 3^n * t^(4*n) * (1 - t^n)^(4*n).
A(t) = 4 at t = 0.291522957845440743948451221223793546169486605337027...
A(t) = 5 at t = 0.329000126460676794243886107787712733546508144420483...
A(t) = 6 at t = 0.357724702510546930008542019234792305087422418573477...
A(t) = 7 at t = 0.380542313167923949301934320871234698370603627590219...
A(t) = 8 at t = 0.398979590930253261958078049297122968769574840959284...
A(t) = 10 at t = 0.42546891301281060442780881746894418736088883964865...
  where 2 = Sum_{n=-oo..+oo} 10^n * t^(4*n) * (1 - t^n)^(4*n).
A(1/3) = 5.13585330720955024782964365699496914153492258811867867...
  where 2 = Sum_{n=-oo..+oo} A(1/3)^n * (3^n - 1)^(4*n) / 3^(4*n*(n+1)).
A(1/4) = 3.17324338350573956221152352367802000913370000075102058...
  where 2 = Sum_{n=-oo..+oo} A(1/4)^n * (4^n - 1)^(4*n) / 4^(4*n*(n+1)).
A(1/5) = 2.44535373134471385251954516592375185067777411156404419...
  where 2 = Sum_{n=-oo..+oo} A(1/5)^n * (5^n - 1)^(4*n) / 5^(4*n*(n+1)).
A(1/6) = 2.07520709570105291415749177247417882089115608856809407...
A(1/7) = 1.85339675365588191953864273073082369078558259828997870...
A(1/8) = 1.70637284932226657507555189122688425270003107631531825...
A(1/9) = 1.60205095825256219436439775831192173599226529828571747...
A(1/10) = 1.5243103888441625451901875329176156768788762565098203...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(sum(m=-#A, #A, x^(4*m) * (1 - x^m +x*O(x^#A))^(4*m) * Ser(A)^m), #A-1); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A357546.

Sequence in context: A009847 A362717 A140226 * A376711 A264924 A008059

Adjacent sequences: A379193 A379194 A379195 * A379197 A379198 A379199

Keyword

nonn

Author

Paul D. Hanna, Jan 14 2025

Status

approved