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A219356 Triangle read by rows: A219274 with rows reversed.

%I A219356 #16 Sep 04 2023 19:31:50
%S A219356 1,1,1,1,2,1,3,1,4,5,1,5,9,16,1,6,14,49,1,7,20,92,70,1,8,27,153,204,
%T A219356 168,1,9,35,235,405,738,768,1,10,44,341,715,1815,3300,1,11,54,474,
%U A219356 1166,3630,9460,7887,1,12,65,637,1794,6578,21307,28743,15015
%N A219356 Triangle read by rows: A219274 with rows reversed.
%C A219356 For more information see A219274.
%H A219356 Alois P. Heinz, <a href="/A219356/b219356.txt">Rows n = 0..100, flattened</a>
%e A219356 A219274 with rows reversed begins:
%e A219356   1;
%e A219356   1;
%e A219356   1;
%e A219356   1,  2;
%e A219356   1,  3;
%e A219356   1,  4,  5;
%e A219356   1,  5,  9,  16;
%e A219356   1,  6, 14,  49;
%e A219356   1,  7, 20,  92,  70;
%e A219356   1,  8, 27, 153, 204, 168;
%e A219356   1,  9, 35, 235, 405, 738, 768;
%e A219356   ...
%p A219356 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A219356       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A219356     end:
%p A219356 g:= proc(n, i, l) local s; s:=i*(i+1)/2;
%p A219356       `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
%p A219356        g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
%p A219356     end:
%p A219356 T:= (n, k)-> `if`(k>n, 0, g(n-k, k-1, [k])):
%p A219356 seq(seq(T(n, n-k), k=0..(n-floor(sqrt(2*n)+1/2))), n=0..14);
%Y A219356 Row lengths are A122797 (for n>0).
%Y A219356 Row sums give: A218293.
%Y A219356 Last elements of rows give: A219339.
%K A219356 nonn,tabf
%O A219356 0,5
%A A219356 _Alois P. Heinz_, Nov 18 2012