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A297359 Array read by antidiagonals: Pascal-like recursion and self-referential boundaries.

%I A297359 #22 Jan 14 2018 14:51:45
%S A297359 1,1,1,1,2,1,1,3,3,1,2,4,6,4,2,1,6,10,10,6,1,1,7,16,20,16,7,1,3,8,23,
%T A297359 36,36,23,8,3,3,11,31,59,72,59,31,11,3,1,14,42,90,131,131,90,42,14,1,
%U A297359 2,15,56,132,221,262,221,132,56,15,2,4,17,71,188,353,483,483,353,188,71,17,4,6,21,88,259,541,836,966,836,541,259
%N A297359 Array read by antidiagonals: Pascal-like recursion and self-referential boundaries.
%C A297359 Array with recursion T(i,j) = T(i-1,j) + T(i,j-1), and boundaries T(0,n) = T(n,0) = a(n). Here a(n) is the array T read by antidiagonals. Require that a(0)=a(1)=1.
%H A297359 Alex Meiburg, <a href="/A297359/b297359.txt">Table of n, a(n) for n = 1..19999</a>
%e A297359 The array looks like
%e A297359   1,  1,  1,   1,   2,    1,    1,    3,    3,   1,   2, ...
%e A297359   1,  2,  3,   4,   6,    7,    8,   11,   14,  15,  17, ...
%e A297359   1,  3,  6,  10,  16,   23,   31,   42,   56,  71,  88, ...
%e A297359   1,  4, 10,  20,  36,   59,   90,  132,  188, 259, 347, ...
%e A297359   2,  6, 16,  36,  72,  131,  221,  353,  541, 800, ...
%e A297359   1,  7, 23,  59, 131,  262,  483,  836, 1377, ...
%e A297359   1,  8, 31,  90, 221,  483,  966, 1802,  ...
%e A297359   3, 11, 42, 132, 353,  836, 1802,  ...
%e A297359   3, 14, 56, 188, 541, 1377,  ...
%e A297359   1, 15, 71, 259, 800,  ...
%e A297359   2, 17, 88, 347, ...
%e A297359 ... [Table corrected and reformatted by _Jon E. Schoenfield_, Jan 14 2018]
%e A297359 The defining property is that when this array is read by antidiagonals we get 1,1,1,1,2,1,... which is both the sequence itself and the top row and first column of the array.
%t A297359 t[a_, b_] := (t[a, b] = t[a, b - 1] + t[a - 1, b]);
%t A297359 t[0, x_] := a[x]; t[x_, 0] := a[x];
%t A297359 a[0] = 1; a[1] = 1;
%t A297359 a[x_] := With[{k = Floor[(Sqrt[8 x + 1] - 1)/2]},
%t A297359   t[x - k (k + 1)/2, (k + 1) (k + 2)/2 - x - 1]]
%t A297359 a /@ Range[60]
%t A297359 TableForm[ Table[t[i, j], {i, 0, 5}, {j, 0, 12}]]
%Y A297359 Cf. A283682, A283681, A283683.
%Y A297359 See also A007318, A297495, A297497, A297188 (antidiagonal sums).
%K A297359 nonn,tabl,easy,nice
%O A297359 1,5
%A A297359 _Alex Meiburg_, Dec 29 2017