A229733 Expansion of (1-x+2*x^3-sqrt(1-2*x-3*x^2+4*x^3-4*x^4))/(2*x^2).

1, 1, 2, 2, 6, 10, 26, 54, 134, 306, 754, 1806, 4478, 11018, 27578, 69014, 174358, 441506, 1124674, 2873182, 7370926, 18962490, 48939530, 126625126, 328475334, 853983058, 2225023890, 5808480046, 15191255262, 39798134122, 104431095898, 274439510838, 722232378998, 1903192922626, 5021490842338, 13264643729406

Offset

0,3

Links

Table of n, a(n) for n=0..35.

S. Kitaev and J. Remmel, The 1-box pattern on pattern avoiding permutations, arXiv preprint arXiv:1305.6970 [math.CO], 2013. See Th. 8.

Formula

D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) -3*(n-1)*a(n-2) +2*(2*n-5)*a(n-3) -4*(n-4)*a(n-4)=0. - R. J. Mathar, Jan 24 2018

a(n) = Sum_{i=0..n+1} Sum_{k=0..n-i} C(i,k)*C(k+i,i)*Sum_{j=0..(-n+3*k+i+2)/2} C(k+1,j)*C(k-j+1,n-2*k+j-i-1)*(-1)^(-n+k+i))/(k+1), a(1)=1. - Vladimir Kruchinin, May 07 2018

Mathematica

CoefficientList[Series[(1 - x + 2 x^3 - Sqrt[1 - 2 x - 3 x^2 + 4 x^3 - 4 x^4]) / (2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, May 30 2019 *)

Prog

(Maxima)
a(n):=if n=1 then 1 else sum(sum((binomial(i, k)*binomial(k+i, i)*(sum(binomial(k+1, j)*binomial(k-j+1, n-2*k+j-i-1), j, 0, (-n+3*k+i+2)/2))*(-1)^(-n+k+i))/(k+1), k, 0, n-i), i, 0, n+1); /* Vladimir Kruchinin, May 07 2018 */

Crossrefs

Sequence in context: A262278 A265639 A208900 * A051765 A265987 A076907

Adjacent sequences: A229730 A229731 A229732 * A229734 A229735 A229736

Keyword

nonn

Author

N. J. A. Sloane, Oct 01 2013

Status

approved