A294159 Alternating row sums of triangle A291844.

1, 1, 2, 6, 18, 55, 171, 538, 1708, 5461, 17560, 56728, 183973, 598597, 1953145, 6388376, 20939664, 68764283, 226192964, 745146462, 2458020664, 8118111977, 26841209903, 88835163150, 294284206183, 975699571009, 3237456793478, 10749922312752, 35718863630895, 118757413662397

Offset: 0

Author

Gheorghe Coserea, Nov 03 2017

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..303

Crossrefs

Cf. A291844.

Programs

PARI

A291843_ser(N, t='t) = {
  my(x='x+O('x^N), y=1, y1=0, n=1,
  dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
  while (n++,
   y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
        (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
   if (y1 == y, break); y = y1; ); y;
};
A291844_ser(N, t='t) = {
  my(z = A291843_ser(N+1, t));
  ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
Vec(A291844_ser(30,-1))
\\ test: y=A291844_ser(200,-1); 0==(x^3 + x^2 + 3*x - 1)*(y^2 - y) + x

Formula

G.f. y(x) satisfies: 0 = (x^3 + x^2 + 3*x - 1)*(y^2 - y) + x.

Conjecture: D-finite with recurrence n*a(n) +(-3*n+1)*a(n-1) +2*(-n+3)*a(n-2) +2*(n-5)*a(n-3) +(n-4)*a(n-4) +(n-5)*a(n-5)=0. - R. J. Mathar, Jun 17 2020