A171416 A sequence with Somos-4 Hankel transform.

1, 1, 2, 3, 7, 13, 31, 65, 156, 351, 849, 1993, 4866, 11733, 28921, 70997, 176560, 438979, 1100302, 2761797, 6969909, 17625015, 44742636, 113822415, 290416803, 742486655, 1902767481, 4885201701, 12567065582, 32382099109, 83580301371

Offset

0,3

Comments

Hankel transform is the Somos-4 sequence A006720(n+2).

The generating function A(x) satisfies A(x) = 1 + x + x^2*A(x) + (x*A(x))^2.

BINOMIAL transform is A087626. HANKEL transform with a(0) omitted is A051138(n+2). - Michael Somos, Jan 11 2013

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

Formula

G.f.: (1 - x^2 - sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2).

G.f.: (1/(1-x))*c(x^2/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.

G.f.: 1/(1-x-x^2/(1-x^2-x^2/(1-x-x^2/(1-x^2-x^2/(1-x-x^2/(1-x^2-x^2/(1-...))))))) (continued fraction).

a(n) = a(n-2) + Sum_{k=1..n-1} a(k-1)*a(n-k-1) with a(0)=a(1)=1.

Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +6*(1-n)*a(n-2) +2*(11-5*n)*a(n-3) +(10-3*n)*a(n-4) +(n-5)*a(n-5)=0. - R. J. Mathar, Jul 24 2012

G.f.: 2*(1 + x) / (1 - x^2 + sqrt(1 - 6*x^2 - 4*x^3 + x^4)).

(n+2) * a(n) - (6*n-6) * a(n-2) - (4*n-10) * a(n-3) + (n-4) * a(n-4) = 0 if n>3. - Michael Somos, Jan 11 2013

If we write the generating function as 1/(1-b_{0}*x/(1-c_{0}x/(1-b_{1}*x/(1-c_{1}*x/(1-...))))), then b_{n}*c_{n} = A006720(n+1)*A006720(n+3)/A006720(n+2)^2 = A377264(n)/A006720(n+2)^2. - Thomas Scheuerle, Oct 22 2024

Example

G.f. = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 13*x^5 + 31*x^6 + 65*x^7 + 156*x^8 + ...

Maple

m:=30; S:=series((1 -x^2 -sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 18 2020

Mathematica

CoefficientList[Series[(1-x^2 -Sqrt[1 -6x^2 -4x^3 +x^4])/(2x^2), {x, 0, 30}], x] (* Or *)
a[n_]:= a[n]= a[n-2] + Sum[a[k-1]a[n-k-1], {k, n-1}]; a[0]=a[1]=1; Array[a, 31, 0] (* Robert G. Wilson v, Mar 28 2011 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 2*(1 + x) / (1 - x^2 + sqrt(1 - 6*x^2 - 4*x^3 + x^4 + x*O(x^n))), n))}; /* Michael Somos, Jan 11 2013 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1-x^2-Sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2))); // G. C. Greubel, Sep 22 2018
(Sage)
def A171416_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1 -x^2 -sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2) ).list()
A171416_list(30) # G. C. Greubel, Feb 18 2020

Crossrefs

Cf. A006720, A051138, A087626, A377264.

Sequence in context: A014234 A124430 A002013 * A193530 A003120 A032131

Adjacent sequences: A171413 A171414 A171415 * A171417 A171418 A171419

Keyword

easy,nonn,changed

Author

Paul Barry, Dec 08 2009

Status

approved