A368773 Antidiagonal sums of A059450.

1, 1, 3, 7, 21, 53, 159, 419, 1257, 3401, 10203, 28095, 84285, 235005, 705015, 1984155, 5952465, 16873745, 50621235, 144327287, 432981861, 1240296773, 3720890319, 10700364691, 32101094073, 92619680089, 277859040267, 803956981807, 2411870945421, 6995553520653, 20986660561959, 61001041404555

Offset: 0

Author

Joerg Arndt, Jan 05 2024

Keywords

• nonn

Crossrefs

Cf. A059450.

Programs

Maple

A368773 := proc(n)
    add(A059450(n-j,j), j=0..floor(n/2)) ;
end proc:
seq(A368773(n),n=0..40) ; # R. J. Mathar, Mar 25 2024

PARI

N=32;  M=matrix(N+1, N+1);  M[1,1] = 1;
T(n,k)= return( M[n+1,k+1] );
{ \\ A059450
 for (n=1, N,
  for (k=0, n,
    v = sum(y=0, n-1, T(y, k) ); \\ vert sum from top
    h = sum(y=0, n-1, T(n, y) ); \\ horiz sum from left
    s = v + h;
    M[ n+1, k+1 ] = s;
    );
); }
\\ antidiagonal sums:
for (n=0, N, my(r=n,c=0, s=0); while( c<=r, s+=T(r,c); r-=1; c+=1 ); print1(s,", "));

Formula

Apparent g.f.: (-b-sqrt(b^2-4*a*c))/(2*a) where a=(6*x^2 - 2*x), b=(-3*x^2 + 4*x - 1), and c=(-x + 1). [determined with Pari's seralgdep()]

Conjecture: D-finite with recurrence +(n+1)*a(n) +3*(-1)*a(n-1) +(-10*n+11)*a(n-2) +3*a(n-3) +9*(n-4)*a(n-4)=0. - R. J. Mathar, Mar 25 2024