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A368773 Antidiagonal sums of A059450.

%I A368773 #17 Jul 09 2024 19:41:23
%S A368773 1,1,3,7,21,53,159,419,1257,3401,10203,28095,84285,235005,705015,
%T A368773 1984155,5952465,16873745,50621235,144327287,432981861,1240296773,
%U A368773 3720890319,10700364691,32101094073,92619680089,277859040267,803956981807,2411870945421,6995553520653,20986660561959,61001041404555
%N A368773 Antidiagonal sums of A059450.
%F A368773 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 A368773 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 A368773 := proc(n)
%p A368773     add(A059450(n-j,j), j=0..floor(n/2)) ;
%p A368773 end proc:
%p A368773 seq(A368773(n),n=0..40) ; # _R. J. Mathar_, Mar 25 2024
%o A368773 (PARI)
%o A368773 N=32;  M=matrix(N+1, N+1);  M[1,1] = 1;
%o A368773 T(n,k)= return( M[n+1,k+1] );
%o A368773 { \\ A059450
%o A368773  for (n=1, N,
%o A368773   for (k=0, n,
%o A368773     v = sum(y=0, n-1, T(y, k) ); \\ vert sum from top
%o A368773     h = sum(y=0, n-1, T(n, y) ); \\ horiz sum from left
%o A368773     s = v + h;
%o A368773     M[ n+1, k+1 ] = s;
%o A368773     );
%o A368773 ); }
%o A368773 \\ antidiagonal sums:
%o A368773 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 A368773 Cf. A059450.
%K A368773 nonn
%O A368773 0,3
%A A368773 _Joerg Arndt_, Jan 05 2024