A276371 Expansion of e.g.f. exp(x/2)/(2 - exp(2*x))^(1/4).

1, 1, 3, 19, 177, 2161, 32643, 587539, 12273537, 291853441, 7782998883, 230028553459, 7462717994097, 263654454838321, 10075889406229923, 414147167601017779, 18217983822073897857, 853975145498805244801, 42495107452208870429763, 2237264405984004517212499, 124243242448367338311920817, 7258224393227482972980320881, 444967879322677755285771182403, 28563002475012109334240250609619

Offset

0,3

Links

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

Formula

E.g.f. A(x) satisfies: A'(x) = A(x)*(1 + A(x)^4)/2 with A(0)=1.

a(2*n) = 0 (mod 3), a(2*n+1) = 1 (mod 3), for n>=0.

a(n) ~ Gamma(3/4) * 2^n * n^(n-1/4) / (sqrt(Pi) * exp(n) * (log(2))^(n+1/4)). - Vaclav Kotesovec, Sep 11 2016

From Seiichi Manyama, Nov 16 2023: (Start)

a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (3/2 * k/n - 2) * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} 2^k * binomial(n-1,k) * a(n-k). (End)

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 177*x^4/4! + 2161*x^5/5! + 32643*x^6/6! + 587539*x^7/7! + 12273537*x^8/8! + 291853441*x^9/9! + 7782998883*x^10/10! +...
such that A(x) = exp(x/2)/(2 - exp(2*x))^(1/4).

Mathematica

With[{nn = 50}, CoefficientList[Series[Exp[x/2]/(2 - Exp[2*x])^(1/4), {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, Apr 09 2017 *)

Prog

(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A = exp(X/2)/(2-exp(2*X))^(1/4); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A124212.

Sequence in context: A305459 A045531 A129481 * A156131 A269421 A304578

Adjacent sequences: A276368 A276369 A276370 * A276372 A276373 A276374

Keyword

nonn

Author

Paul D. Hanna, Sep 09 2016

Status

approved