A207319 a(n) = Sum_{k=0..n-1} (-1)^k*k*A000172(k).

0, 0, -2, 18, -150, 1234, -10026, 81078, -653642, 5259654, -42268734, 339383866, -2723191254, 21839901546, -175088593334, 1403249333706, -11243655022134, 90073138966378, -721461395853162, 5777933202466686, -46268213346433298, 370468327360598622, -2966084621457856170, 23745699326099825110

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. 1.6.

Programs

Mathematica

a[n_] := Sum[(-1)^k k Sum[Binomial[k, j]^3, {j, 0, k}], {k, 0, n-1}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 31 2018 *)

Formula

Conjecture: (n-1)*(n-2)*a(n) +2*(n^2+3*n-9)*a(n-1) +3*(-13*n^2+63*n-80)*a(n-2) +4*(17*n^2-96*n+136)*a(n-3) -32*(n-3)^2*a(n-4)=0. - R. J. Mathar, Nov 28 2013