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

Offset

0,2

Comments

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. - Michael Somos, May 26 2022

Links

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

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

a(n) = (-2)^n * (P_n(-2) - P_{n-1}(-2))/2 if n>0 where P_n(x) is Legendre polynomial. - Michael Somos, May 26 2022

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)

Example

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

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 *)

Prog

(Maxima)
a(n):=sum(2^(n-i)*binomial(n, i)*binomial(n+i-1, i), i, 0, n);
(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
(PARI) 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

Crossrefs

Cf. A110170.

Sequence in context: A074550 A205576 A173695 * A370186 A361843 A097188

Adjacent sequences: A255685 A255686 A255687 * A255689 A255690 A255691

Keyword

nonn,easy

Author

Vladimir Kruchinin, Mar 15 2015

Status

approved