cp's OEIS Frontend

A350519 a(n) = A(n,n) where A(1,n) = A(n,1) = prime(n+1) and A(m,n) = A(m-1,n) + A(m,n-1) + A(m-1,n-1) for m > 1 and n > 1.

%I A350519 #28 Apr 22 2025 19:11:59
%S A350519 3,13,63,325,1719,9237,50199,275149,1518263,8422961,46935819,
%T A350519 262512929,1472854451,8285893713,46723439019,264009961733,
%U A350519 1494486641911,8473508472009,48112827862527,273541139290857,1557023508876891,8872219429659729,50605041681538595,288897992799897481
%N A350519 a(n) = A(n,n) where A(1,n) = A(n,1) = prime(n+1) and A(m,n) = A(m-1,n) + A(m,n-1) + A(m-1,n-1) for m > 1 and n > 1.
%C A350519 Replacing prime(n+1) by other functions f(n) we can get many other sequences. For example, with  f(n) = 1 we get A001850.
%e A350519 The two-dimensional recurrence A(m,n) can be depicted in matrix form as
%e A350519    3   5   7   11   13    17    19 ...
%e A350519    5  13  25   43   67    97   133 ...
%e A350519    7  25  63  131  241   405   635 ...
%e A350519   11  43 131  325  697  1343  2383 ...
%e A350519   13  67 241  697 1719  3759  7485 ...
%e A350519   17  97 405 1343 3759  9237 20481 ...
%e A350519   19 133 635 2383 7485 20481 50199 ...
%e A350519   ...
%e A350519 and then a(n) is the main diagonal of this matrix, A(n,n).
%t A350519 f[1,1]=3;f[m_,1]:=Prime[m+1];f[1,n_]:=Prime[n+1];f[m_,n_]:=f[m,n]=f[m-1,n]+f[m,n-1]+f[m-1,n-1];Table[f[n,n],{n,25}] (* _Giorgos Kalogeropoulos_, Jan 03 2022 *)
%o A350519 (MATLAB)
%o A350519 clear all
%o A350519 close all
%o A350519 sz = 14
%o A350519 f = zeros(sz,sz);
%o A350519 pp = primes(50);
%o A350519 f(1,:) = pp(2:end);
%o A350519 f(:,1) = pp(2:end);
%o A350519 for m=2:sz
%o A350519     for  n=2:sz
%o A350519         f(m,n) = f(m-1,n-1)+f(m,n-1)+f(m-1,n);
%o A350519     end
%o A350519 end
%o A350519 an = []
%o A350519 for n=1:sz
%o A350519     an = [an f(n,n)];
%o A350519 end
%o A350519 S = sprintf('%i,',an);
%o A350519 S = S(1:end-1)
%Y A350519 Cf. A000040, A001850, A002002, A050151, A344576 (see comments).
%K A350519 nonn
%O A350519 1,1
%A A350519 _Yigit Oktar_, Jan 02 2022