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A345372 a(n) = Sum_{i=1..n} nac(i,n) where nac(i,n) is the n-th i-bonacci number. The n-th i-bonacci number here is equal to 1 for the first i terms, with subsequent terms equaling the sum of the previous n terms.

%I A345372 #43 May 29 2022 12:45:53
%S A345372 1,2,4,8,16,31,60,114,217,411,780,1481,2820,5379,10288,19720,37884,
%T A345372 72924,140640,271695,525698,1018611,1976276,3838889,7465191,14531683,
%U A345372 28313776,55214993,107762464,210477611,411387724,804609206,1574671586,3083549861,6041628460
%N A345372 a(n) = Sum_{i=1..n} nac(i,n) where nac(i,n) is the n-th i-bonacci number. The n-th i-bonacci number here is equal to 1 for the first i terms, with subsequent terms equaling the sum of the previous n terms.
%C A345372 a(n) is the sum of the first n elements of the n-th column of the following array:
%C A345372   1, 1, 1, 1, 1, ... (1-bonacci numbers)
%C A345372   1, 1, 2, 3, 5, ... (2-bonacci or Fibonacci numbers)
%C A345372   1, 1, 1, 3, 5, ... (3-bonacci or tribonacci numbers)
%C A345372   1, 1, 1, 1, 4, ... (4-bonacci or tetranacci numbers)
%C A345372   ...
%C A345372 For n >= 3, this sequence is 2 + antidiagonal sums of A061451.
%F A345372 a(n) = Sum_{i=1..n} nac(i,n) where nac(i,n) = 1 if 1 <= n <= i, Sum_{k=1..i} nac(i,n-k) if n > i.
%p A345372 b:= proc(i, n) option remember; `if`(n=0, 0,
%p A345372       `if`(n<=i, 1, add(b(i, n-j), j=1..i)))
%p A345372     end:
%p A345372 a:= n-> add(b(i, n), i=1..n):
%p A345372 seq(a(n), n=1..36);  # _Alois P. Heinz_, Jun 16 2021
%t A345372 b[i_, n_] := b[i, n] = If[n==0, 0,
%t A345372      If[n<=i, 1, Sum[b[i, n-j], {j, 1, i}]]];
%t A345372 a[n_] := Sum[b[i, n], {i, 1, n}];
%t A345372 Table[a[n], {n, 1, 36}] (* _Jean-François Alcover_, May 29 2022, after _Alois P. Heinz_ *)
%Y A345372 Cf. A000045, A000213, A000288, A000322, A000383, A060455, A061451.
%K A345372 nonn
%O A345372 1,2
%A A345372 _Christoph B. Kassir_, Jun 16 2021