A178115 -id:A178115 - cp's OEIS frontend

A178113 Transform of C(n+1,floor((n+1)/2)) by A178112.

1, 2, 2, 4, 5, 10, 13, 26, 35, 70, 96, 192, 267, 534, 750, 1500, 2123, 4246, 6046, 12092, 17303, 34606, 49721, 99442, 143365, 286730, 414584, 829168, 1201917, 2403834, 3492117, 6984234, 10165779, 20331558, 29643870, 59287740, 86574831, 173149662

Offset: 0

Paul Barry, May 20 2010

Hankel transform is A178115.

Cf. A178112, A178115, A005773.

a(n) = Sum_{k=0..n} C(floor(n/2),floor(k/2))*((1+(-1)^(n-k))/2)*C(k+1,floor((k+1)/2)). [The formula seems to generate A026392, not these terms. - R. J. Mathar, Feb 10 2015]

Conjecture: a(2*n) = A005773(n+1), a(2*n+1) = 2*a(2*n). - Jason Yuen, Feb 09 2025

A178114 Expansion of (sqrt(1-2x+7x^2-6x^3+5x^4)-(1-x+x^2))/(2x^2(1-x+x^2)).

Original entry on oeis.org

1, 1, -1, -3, 0, 8, 6, -21, -37, 45, 175, -22, -703, -533, 2370, 4321, -5930, -23560, 3534, 104035, 81083, -376267, -705158, 993738, 4047745, -604007, -18622243, -14895477, 69622834, 133284470, -188549209, -784970693, 110402283

Offset: 0

Paul Barry, May 20 2010

Hankel transform is A178115.

CoefficientList[Series[(Sqrt[1-2x+7x^2-6x^3+5x^4]-(1-x+x^2))/(2x^2 (1-x+x^2)),{x,0,40}],x] (* Harvey P. Dale, Dec 01 2013 *)

D-finite with recurrence: +(n+2)*a(n) +2*(-n-1)*a(n-1) +(7*n-4)*a(n-2) +2*(-3*n+4)*a(n-3) +5*(n-2)*a(n-4)=0. - R. J. Mathar, Feb 10 2015

cp's OEIS Frontend

A178113 Transform of C(n+1,floor((n+1)/2)) by A178112.

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