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
Examples
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 + ...
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.
Programs
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Magma
R
:=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 -
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
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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 *) -
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 */ -
Sage
def A171416_list(prec): P.
= 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
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
Comments