A113682 Expansion of 2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2))).

1, 1, 4, 9, 26, 70, 197, 553, 1570, 4476, 12827, 36894, 106471, 308113, 893804, 2598313, 7567466, 22076404, 64498427, 188689684, 552675365, 1620567763, 4756614062, 13974168190, 41088418151, 120906613075, 356035078102

Offset: 0

Author

Paul Barry, Nov 04 2005

Keywords

• easy
• nonn

Comments

Convolution of A002426 and A005043. Diagonal sums of A094531.

Hankel transform is A164611. - Paul Barry, Aug 17 2009

David Scambler observed that [1,0,a(n-2)] for n>=2 count the Dyck paths of semilength n such that the number of peaks equals the number of hills plus the number of returns. - Peter Luschny, Oct 22 2012

Conjectural congruences (working with an offset of 1): a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. - Peter Bala, Mar 15 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Crossrefs

Cf. A002426, A005043, A094531, A217539, A217540, A246437, A005717.

Programs

Magma

[(Evaluate(GegenbauerPolynomial(n+1, -n-1), -1/2) + (-1)^n)/2: n in [0..40]];  // G. C. Greubel, Apr 04 2024

Mathematica

ex[x_]:=Module[{sx=Sqrt[1-2x-3x^2]},2/(sx (1+x+sx))]; CoefficientList[ Series[ ex[x],{x,0,40}],x] (* Harvey P. Dale, May 28 2012 *)
Flatten[{1, Table[Coefficient[Sum[(1 + x + x^2)^k, {k, 0, n}], x^n], {n, 1, 30}]}] (* Vaclav Kotesovec, Jan 08 2016 *)

Maxima

makelist((ultraspherical(n+1,-n-1,-1/2)+(-1)^n)/2,n,0,12); /* Emanuele Munarini, Dec 20 2016 */

PARI

x='x+O('x^50); Vec(2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2)))) \\ G. C. Greubel, Feb 28 2017

SageMath

[(gegenbauer(n+1,-n-1,-1/2) +(-1)^n)/2 for n in range(41)] # G. C. Greubel, Apr 04 2024

Formula

a(n) = Sum_{k=0..floor(n/2)} ( Sum_{i=0..n-k} C(n-2k-i, i)*C(n-k, k+i) ).

a(n) = Sum_{k=0..n} A002426(k)*A005043(n-k).

a(n) = Sum_{k=0..n} C(n+1,k+1)*C(k,n-k). - Paul Barry, Aug 21 2007

a(n) = (A002426(n+1) + (-1)^n)/2. - Paul Barry, Aug 17 2009

G.f.: d/dx log(1/(1-x*A005043(x))). - Vladimir Kruchinin, Apr 18 2011

D-finite with recurrence: (n+1)*a(n) +(-n-1)*a(n-1) +(-5*n+1)*a(n-2) +3*(-n+1)*a(n-3)=0. - R. J. Mathar, Nov 26 2012

Recurrence: (n+4)*a(n+3)-(n+4)*a(n+2)-(5*n+14)*a(n+1)-3*(n+2)*a(n)=0. Remark: this recurrence can be obtained using the identity a(n) = (t(n+1)+(-1)^n)/2 and the recurrence of the central trinomial coefficients t(n) = A002426(n). So, the above P-finite recurrences are true. - Emanuele Munarini, Dec 20 2016

a(n) = (-1)^(n+1) * (hypergeom([1/2, -n-1], [1], 4) - 1)/2. - Vladimir Reshetnikov, Apr 25 2016

a(n) = (-1)^n + A246437(n+1). - Vladimir Reshetnikov, Apr 25 2016