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%I #2 Mar 30 2012 18:37:15
%S 1,1,1,3,2,1,14,8,3,1,86,45,14,4,1,645,318,86,22,5,1,5662,2671,645,
%T 152,31,6,1,56632,25805,5662,1251,232,41,7,1,633545,280609,56632,
%U 11869,2026,327,53,8,1,7820115,3381993,633545,126987,20143,2991,457,66,9,1
%N Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {m*(m+1)/2, m>=0} and then taking partial sums, starting with all 1's in row 0.
%e Table begins:
%e (1),(1),1,(1),1,1,(1),1,1,1,(1),1,1,1,1,(1),1,...;
%e (1),(2),3,(4),5,6,(7),8,9,10,(11),12,13,14,15,(16),...;
%e (3),(8),14,(22),31,41,(53),66,80,95,(112),130,149,169,190,...;
%e (14),(45),86,(152),232,327,(457),606,775,965,(1202),1464,1752,2067,...;
%e (86),(318),645,(1251),2026,2991,(4455),6207,8274,10684,(13934),17653,...;
%e (645),(2671),5662,(11869),20143,30827,(48480),70355,96990,128959,...;
%e (5662),(25805),56632,(126987),223977,352936,(582183),874664,1240239,...;
%e (56632),(280609),633545,(1508209),2748448,4438122,(7641111),11831184,...;
%e (633545),(3381993),7820115,(19651299),36837937,60743909,...; ...
%e where row n equals the partial sums of row n-1 after removing terms
%e at positions {m*(m+1)/2, m>=0} (marked by parenthesis in above table).
%e For example, to generate row 3 from row 2:
%e [3,8, 14, 22, 31,41, 53, 66,80,95, 112, 130,...]
%e remove terms at positions {0,1,3,6,10,...}, yielding:
%e [14, 31,41, 66,80,95, 130,149,169,190, ...]
%e then take partial sums to obtain row 3:
%e [14, 45,86, 152,232,327, 457,606,775,965, ...].
%e Continuing in this way generates all rows of this table.
%e RELATION TO POWERS OF A SPECIAL TRIANGULAR MATRIX.
%e Columns 0 and 1 are found in triangle T=A152400, which begins:
%e 1;
%e 1, 1;
%e 3, 2, 1;
%e 14, 8, 3, 1;
%e 86, 45, 15, 4, 1;
%e 645, 318, 99, 24, 5, 1;
%e 5662, 2671, 794, 182, 35, 6, 1;
%e 56632, 25805, 7414, 1636, 300, 48, 7, 1; ...
%e where column k of T = column 0 of matrix power T^(k+1) for k>=0.
%e Furthermore, matrix powers of triangle T=A152400 satisfy:
%e column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.
%e Column 3 of this square array = column 1 of T^2:
%e 1;
%e 2, 1;
%e 8, 4, 1;
%e 45, 22, 6, 1;
%e 318, 152, 42, 8, 1;
%e 2671, 1251, 345, 68, 10, 1;
%e 25805, 11869, 3253, 648, 100, 12, 1; ...
%e RELATED TRIANGLE A127714 begins:
%e 1;
%e 1, 1, 1;
%e 1, 2, 2, 3, 3, 3;
%e 1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
%e 1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...
%e where right border = column 0 of this square array.
%o (PARI) {T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==m*(m+1)/2, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}
%Y Cf. columns: A127715, A152401, A152404.
%Y Cf. related triangles: A152400, A127714.
%Y Cf. variants: A125781, A127054, A136212, A136217, A135876, A135878, A136730.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Dec 05 2008
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