A098777 -id:A098777 - cp's OEIS frontend

A144689 A098777 mod 7.

1, 6, 5, 2, 2, 2, 2, 4, 1, 6, 6, 6, 6, 5, 3, 4, 4, 4, 4, 1, 2, 5, 5, 5, 5, 3, 6, 1, 1, 1, 1, 2, 4, 3, 3, 3, 3, 6, 5, 2, 2, 2, 2, 4, 1, 6, 6, 6, 6, 5, 3, 4, 4, 4, 4, 1, 2, 5, 5, 5, 5, 3, 6, 1, 1, 1, 1, 2, 4, 3, 3, 3, 3, 6, 5, 2, 2, 2, 2, 4, 1, 6, 6, 6, 6, 5, 3, 4, 4, 4, 4, 1, 2, 5, 5, 5, 5, 3, 6, 1, 1

Offset: 0

N. J. A. Sloane, Feb 08 2009

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..1000
R. Bacher and P. Flajolet, Pseudo-factorials, Elliptic Functions and Continued Fractions, arXiv:0901.1379 [math.CA], 2009.

a:= proc(n) option remember; `if`(n=0,1,(-1)^n*add(binomial(n-1,k)*a(k)*a(n-1-k),k=0..n-1)) end: seq(modp(a(n),7), n=0..100); # Muniru A Asiru, Jul 29 2018

b[0] = 1; b[n_] := b[n] = (-1)^n Sum[Binomial[n-1, k] b[k] b[n-k-1], {k, 0, n-1}];
a[n_] := Mod[b[n], 7]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 29 2018 *)

For n >= 0 has period 36.
From Chai Wah Wu, Jun 09 2016: (Start)
a(n) = a(n-1) - a(n-18) + a(n-19) for n > 19.
G.f.: (1 + 5*x - x^2 - 3*x^3 + 2*x^7 - 3*x^8 + 5*x^9 - x^13 - 2*x^14 + x^15 + x^18 + 2*x^19)/(1 - x + x^18 - x^19). (End)

A144750 A098777 mod 9.

Original entry on oeis.org

1, 8, 7, 2, 7, 5, 4, 5, 1, 8, 1, 2, 7, 2, 4, 5, 4, 8, 1, 8, 7, 2, 7, 5, 4, 5, 1, 8, 1, 2, 7, 2, 4, 5, 4, 8, 1, 8, 7, 2, 7, 5, 4, 5, 1, 8, 1, 2, 7, 2, 4, 5, 4, 8, 1, 8, 7, 2, 7, 5, 4, 5, 1, 8, 1, 2, 7, 2, 4, 5, 4, 8, 1, 8, 7, 2, 7, 5, 4, 5, 1, 8, 1, 2, 7, 2, 4, 5, 4, 8, 1, 8, 7, 2, 7, 5, 4, 5, 1, 8, 1

Offset: 0

N. J. A. Sloane, Feb 08 2009

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..1000
R. Bacher and P. Flajolet, Pseudo-factorials, Elliptic Functions and Continued Fractions, arXiv:0901.1379 [math.CA], 2009.

a:= proc(n) option remember; `if`(n=0,1,(-1)^n*add(binomial(n-1,k)*a(k)*a(n-1-k),k=0..n-1)) end: seq(modp(a(n),9), n=0..100); # Muniru A Asiru, Jul 29 2018

b[0] = 1;
b[n_] := b[n] = (-1)^n Sum[Binomial[n-1, k] b[k] b[n-k-1], {k, 0, n-1}];
a[n_] := Mod[b[n], 9]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 29 2018 *)

From Chai Wah Wu, Nov 30 2018: (Start)
a(n) = a(n-2) + a(n-3) - a(n-5) - a(n-6) + a(n-8) for n > 7 (conjectured).
G.f.: (-8*x^7 - 4*x^6 + 3*x^5 + 8*x^4 + 7*x^3 - 6*x^2 - 8*x - 1)/((x - 1)*(x + 1)*(x^6 - x^3 + 1)) (conjectured). (End)

A235166 E.g.f. satisfies: A'(x) = A(x)^2/A(-x)^2, with A(0)=1.

Original entry on oeis.org

1, 1, 4, 16, 88, 640, 5440, 54400, 620800, 7966720, 113651200, 1783091200, 30519808000, 565916876800, 11300689100800, 241781039104000, 5517822373888000, 133795711025152000, 3435107208822784000, 93093522064998400000, 2655675672405606400000, 79546285618254315520000

Offset: 0

Paul D. Hanna, Jan 04 2014

nonn

See comments by Roland Bacher in A098777 which imply that this sequence is related to elliptic functions.

Compare to: G'(x) = G(x)^2/G(-x) dx, which holds when G(x) = 1/(cos(x) - sin(x)), the e.g.f. of A001586 (Springer numbers).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 88*x^4/4! + 640*x^5/5! +...
Related series.
A(x)^2 = 1 + 2*x + 10*x^2/2! + 56*x^3/3! + 400*x^4/4! + 3440*x^5/5! +...
1/A(x) = 1 - x - 2*x^2/2! + 2*x^3/3! + 16*x^4/4! - 40*x^5/5! - 320*x^6/6! +...+ A098777(n)*x^n/n! +...

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (first 100 terms from Paul D. Hanna)

Cf. A001586, A098777.

kmax = 21;
A[x_] = 1+x; Do[A[x_] = 1+Integrate[A[x]^2/A[-x]^2+O[x]^k, x] // Normal, {k, 1, kmax}];
CoefficientList[A[x], x] Range[0, kmax]! (* Jean-François Alcover, Jul 29 2018 *)

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(A^2/subst(A, x,-x +x*O(x^n))^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

{a(n)=local(A=1); A=1/(1-3*serreverse(intformal(1/(1-9*x^2 +x*O(x^n))^(2/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: 1/(1 - 3*Series_Reversion( Integral 1/(1 - 9*x^2)^(2/3) dx ))^(1/3).
E.g.f.: 1/F(x), where F(x) equals the e.g.f. of A098777 (pseudo-factorials).
a(n) ~ 2^(-2/3) * n! * (9*GAMMA(2/3)^3/(2^(2/3)*Pi^2))^(n+1). - Vaclav Kotesovec, Feb 24 2014

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A144689 A098777 mod 7.

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A144750 A098777 mod 9.

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A235166 E.g.f. satisfies: A'(x) = A(x)^2/A(-x)^2, with A(0)=1.

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