A207318 a(n) = Sum_{k=0..n-1} (-1)^k*A000172(k).

0, 1, -1, 9, -47, 299, -1953, 13231, -91729, 647433, -4633499, 33531761, -244884159, 1802040241, -13346305519, 99392117841, -743734839215, 5588564785067, -42148760792553, 318928716891883, -2420342154102853, 18416484881248743, -140466988872011009, 1073705008744247231, -8223501739695527745

Offset: 0

Author

N. J. A. Sloane, Feb 16 2012

Keywords

• sign

Links

• Z.-W. Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034, 2011. See Eq. 1.5.

Programs

Mathematica

Flatten[{0,Table[Sum[(-1)^k*Sum[Binomial[k,j]^3,{j,0,k}],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 31 2014 *)

Formula

Conjecture: (n-1)^2*a(n) +(2*n-3)*(3*n-5)*a(n-1) +(-15*n^2+53*n-48)*a(n-2) +8*(n-2)^2*a(n-3)=0. - R. J. Mathar, Nov 28 2013

a(n) ~ (-1)^(n+1) * sqrt(3) * 2^(3*n+1) / (27*Pi*n). - Vaclav Kotesovec, Jan 31 2014