A255688 G.f.: (2*x+1)/(2*sqrt(4*x^2-8*x+1)) + 1/2.
1, 3, 15, 90, 579, 3858, 26262, 181380, 1265955, 8906706, 63058530, 448716876, 3206387790, 22992276180, 165364807308, 1192393813320, 8617219956003, 62397513984210, 452607991376490, 3288138397237884, 23921128800374874, 174244027232971548, 1270655996584434420
Offset: 0
Examples
G.f. = 1 + 3*x + 15*x^2 + 90*x^3 + 579*x^4 + 3858*x^5 + 26262*x^6 + ... - _Michael Somos_, May 26 2022
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
CoefficientList[Series[(2*x+1)/(2*Sqrt[4*x^2-8*x+1])+1/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 15 2015 *) -
Maxima
a(n):=sum(2^(n-i)*binomial(n,i)*binomial(n+i-1,i),i,0,n);
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PARI
my(x='x+O('x^50)); Vec((2*x+1)/(2*sqrt(4*x^2-8*x+1)) + 1/2) \\ G. C. Greubel, Jun 03 2017 -
Sage
a = lambda n: 2^n*hypergeometric([-n, n], [1], -1/2).simplify() [a(n) for n in range(21)] # Peter Luschny, Mar 15 2015
Formula
a(n) = Sum_{i=0..n} 2^(n-i)*binomial(n,i)*binomial(n+i-1,i).
a(n) ~ 3^(1/4) * 2^(n-1) * (2+sqrt(3))^n / sqrt(Pi*n). - Vaclav Kotesovec, Mar 15 2015
a(n) = 2^n*hypergeom([-n, n], [1], -1/2). - Peter Luschny, Mar 15 2015
D-finite with recurrence: n*a(n) -6*n*a(n-1) +12*(-n+3)*a(n-2) +8*(n-3)*a(n-3)=0. - R. J. Mathar, Jan 25 2020
From Michael Somos, May 26 2022: (Start)
a(n) = (-2)^n * (P_n(-2) - P_{n-1}(-2))/2 if n>0 where P_n(x) is Legendre polynomial.
0 = a(n)*(+64*a(n+1) -96*a(n+2) -192*a(n+3) +32*a(n+4)) +a(n+1)*(-96*a(n+1) +240*a(n+2) +272*a(n+3) -48*a(n+4)) +a(n+2)*(-144*a(n+2) +60*a(n+3) -6*a(n+4)) + a(n+3)*(-6*a(n+3) +a(n+4)) if n>=0. (End)
From Peter Bala, Nov 08 2022: (Start)
a(n) = [x^n] ( (1 + 2*x)/(1 - x) )^n.
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p and all positive integers n and r. (End)
a(n) = 3*A098663(n-1) = Sum_{k = 0..n-1} binomial(n-1, k)*binomial(n, k+1)* 3^(k+1) for n >= 1. - Peter Bala, Sep 02 2024