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%I #8 Apr 28 2016 11:36:46
%S 1,1,1,1,2,1,2,1,3,4,6,1,3,4,6,1,4,8,14,15,18,22,28,1,4,8,14,15,18,22,
%T 28,1,5,13,27,42,60,82,110,111,115,123,137,152,170,192,220,1,5,13,27,
%U 42,60,82,110,111,115,123,137,152,170,192,220,1,6,19,46,88,148,230,340,451,566,689,826,978,1148,1340,1560,1561,1566,1579,1606,1648,1708,1790,1900,2011,2126,2249,2386,2538,2708,2900,3120
%N Triangle, read by rows consisting of terms T(n,k), k=0..2^n-1, where row n+1 equals the concatenation of the partial sums of the prior row with itself.
%H Paul D. Hanna, <a href="/A256600/b256600.txt">Table of n, a(n) for n = 0..16382</a>
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1, 2;
%e 1, 3, 4, 6, 1, 3, 4, 6;
%e 1, 4, 8, 14, 15, 18, 22, 28, 1, 4, 8, 14, 15, 18, 22, 28;
%e 1, 5, 13, 27, 42, 60, 82, 110, 111, 115, 123, 137, 152, 170, 192, 220, 1, 5, 13, 27, 42, 60, 82, 110, 111, 115, 123, 137, 152, 170, 192, 220;
%e 1, 6, 19, 46, 88, 148, 230, 340, 451, 566, 689, 826, 978, 1148, 1340, 1560, 1561, 1566, 1579, 1606, 1648, 1708, 1790, 1900, 2011, 2126, 2249, 2386, 2538, 2708, 2900, 3120, 1, 6, 19, 46, 88, 148, 230, 340, 451, 566, 689, 826, 978, 1148, 1340, 1560, 1561, 1566, 1579, 1606, 1648, 1708, 1790, 1900, 2011, 2126, 2249, 2386, 2538, 2708, 2900, 3120; ...
%e Illustration of generating method.
%e Given row 2: [1, 2, 1, 2],
%e take partial sums: [1, 3, 4, 6],
%e then concatenate with itself to form row 3:
%e [1, 3, 4, 6, 1, 3, 4, 6].
%e Continuing in this way will generate all the rows of the triangle.
%e The final terms in each row form sequence A256599:
%e [1, 1, 2, 6, 28, 220, 3120, 83664, 4357344, 447134112, 91076016768, ...].
%o (PARI) {A=[1];for(n=1,8,print(A);A=concat(Vec(Ser(A)/(1-x)),Vec(Ser(A)/(1-x)));)}
%Y Cf. A256599.
%K nonn,tabf
%O 0,5
%A _Paul D. Hanna_, Apr 03 2015
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