A368773 - OEIS

%I #17 Jul 09 2024 19:41:23

%S 1,1,3,7,21,53,159,419,1257,3401,10203,28095,84285,235005,705015,

%T 1984155,5952465,16873745,50621235,144327287,432981861,1240296773,

%U 3720890319,10700364691,32101094073,92619680089,277859040267,803956981807,2411870945421,6995553520653,20986660561959,61001041404555

%N Antidiagonal sums of A059450.

%F 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()]

%F 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

%p A368773 := proc(n)

%p     add(A059450(n-j,j), j=0..floor(n/2)) ;

%p end proc:

%p seq(A368773(n),n=0..40) ; # _R. J. Mathar_, Mar 25 2024

%o (PARI)

%o N=32;  M=matrix(N+1, N+1);  M[1,1] = 1;

%o T(n,k)= return( M[n+1,k+1] );

%o { \\ A059450

%o  for (n=1, N,

%o   for (k=0, n,

%o     v = sum(y=0, n-1, T(y, k) ); \\ vert sum from top

%o     h = sum(y=0, n-1, T(n, y) ); \\ horiz sum from left

%o     s = v + h;

%o     M[ n+1, k+1 ] = s;

%o     );

%o ); }

%o \\ antidiagonal sums:

%o for (n=0, N, my(r=n,c=0, s=0); while( c<=r, s+=T(r,c); r-=1; c+=1 ); print1(s,", "));

%Y Cf. A059450.

%K nonn

%O 0,3

%A _Joerg Arndt_, Jan 05 2024