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%I #21 Jul 09 2021 13:01:43
%S 1,3,5,2,7,4,4,9,6,8,4,4,11,8,12,8,12,0,8,4,13,10,16,12,20,4,16,8,8,4,
%T 8,4,4,15,12,20,16,28,8,28,12,16,8,24,8,16,8,4,4,8,8,8,0,4,17,14,24,
%U 20,36,12,40,20,24,12,40,12,36,16,8,16,28,16,24,4,8,8,16,4,12,8,8,0,12,4,8,0,0,4
%N T(n,k) is the number of k's in the n-th row of Stern's triangle (A337277); triangle T(n,k), n >= 0, 1 <= k <= A000045(n+1), read by rows.
%C All terms in the first column are odd, all other terms are even.
%H Alois P. Heinz, <a href="/A334627/b334627.txt">Rows n = 0..21, flattened</a>
%F Sum_{k=1..A000045(n+1)} k * T(n,k) = A000244(n).
%e T(0,1) = 1 because Stern's triangle has one 1 in row n=0.
%e T(2,2) = 2 because Stern's triangle has two 2's in row n=2.
%e T(4,3) = 8 because Stern's triangle has eight 3's in row n=4.
%e Triangle T(n,k) begins:
%e 1;
%e 3;
%e 5, 2;
%e 7, 4, 4;
%e 9, 6, 8, 4, 4;
%e 11, 8, 12, 8, 12, 0, 8, 4;
%e 13, 10, 16, 12, 20, 4, 16, 8, 8, 4, 8, 4, 4;
%e 15, 12, 20, 16, 28, 8, 28, 12, 16, 8, 24, 8, 16, 8, 4, 4, 8, 8, 8, 0, 4;
%e ...
%p b:= proc(n) option remember; `if`(n=0, [1], (l-> [1, l[1],
%p seq([l[i-1]+l[i], l[i]][], i=2..nops(l)), 1])(b(n-1)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(x^j, j=b(n))):
%p seq(T(n), n=0..8);
%Y Column k=1 gives A005408.
%Y Row sums give A126646.
%Y Row lengths give A000045(n+1).
%Y Cf. A000244, A337277.
%K nonn,look,tabf
%O 0,2
%A _Alois P. Heinz_, Sep 09 2020
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