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A113682
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Expansion of 2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2))).
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8
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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} C(n+1,k+1)*C(k,n-k). - Paul Barry, Aug 21 2007
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
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MATHEMATICA
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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 *)
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PROG
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(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
(Magma) [(Evaluate(GegenbauerPolynomial(n+1, -n-1), -1/2) + (-1)^n)/2: n in [0..40]]; // G. C. Greubel, Apr 04 2024
(SageMath) [(gegenbauer(n+1, -n-1, -1/2) +(-1)^n)/2 for n in range(41)] # G. C. Greubel, Apr 04 2024
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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