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%I #3 Mar 30 2012 18:37:02
%S 1,1,1,1,2,1,1,4,3,1,1,9,8,4,1,1,24,23,14,5,1,1,77,76,52,21,6,1,1,295,
%T 294,217,91,29,7,1,1,1329,1328,1033,433,141,39,8,1,1,6934,6933,5604,
%U 2307,739,216,50,9,1,1,41351,41350,34416,13804,4276,1274,306,62,10,1,1
%N Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 - 2 (m>=2).
%C Generated by a method similar to Moessner's factorial triangle (A125714).
%F Surprisingly, column 1 equals A091352 = column 1 of triangle A091351, in which column k equals row sums of the matrix power A091351^k. Column 3 of this table also equals column 1 of matrix power A091351^2.
%e Rows are partial sums excluding terms in columns k = {1,4,8,13,...}:
%e row 2 = partial sums of [1, 3,4, 6,7,8, 10,11,12,13, ...];
%e row 3 = partial sums of [1, 8,14, 29,39,50, 75,90,106,123, ...];
%e row 4 = partial sums of [1, 23,52, 141,216,306, 535,695,876,1079,...].
%e The terms that are excluded in the partial sums are shown enclosed in
%e parenthesis in the table below. Rows of this table begin:
%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, 17, ...;
%e 1,(4), 8, 14,(21), 29, 39, 50,(62), 75, 90, 106, 123,(141), 160, 181,.;
%e 1,(9), 23, 52,(91), 141, 216, 306,(412), 535, 695, 876, 1079,(1305),..;
%e 1,(24), 76, 217,(433), 739, 1274, 1969,(2845), 3924, 5479, 7335,...;
%e 1,(77), 294, 1033,(2307), 4276, 8200, 13679,(21014), 30534, 45528,...;
%e 1,(295), 1328, 5604,(13804), 27483, 58017, 103545,(167868), 255305,...;
%e 1,(1329), 6933, 34416,(92433), 195978, 451283, 855463,(1454823),...;
%e 1,(6934), 41350, 237328,(688611), 1544074, 3847960, 7700971,...;
%e 1,(41351), 278679, 1822753,(5670713), 13371684, 35818351, 75299744,...;
%e 1,(278680), 2101433, 15473117,(51291468), 126591212, 362337006,...;
%e 1,(2101434), 17574551, 144165763,(506502769), 1303252476,...;
%e 1,(17574552), 161740315, 1464992791,(5430460072), 14517950305,...;
%e Column 1 of this table equals column 1 of triangle A091351;
%e triangle A091351 begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 4, 3, 1;
%e 1, 9, 9, 4, 1;
%e 1, 24, 30, 16, 5, 1;
%e 1, 77, 115, 70, 25, 6, 1;
%e 1, 295, 510, 344, 135, 36, 7, 1;
%e 1, 1329, 2602, 1908, 805, 231, 49, 8, 1; ...
%e where column k of A091351 = row sums of matrix power A091351^k for k>=0.
%o (PARI) {T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b+1)/2-2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}
%Y Cf. A091351, A091352; columns: A125782, A125783, A125784, A125785, A125786; diagonals: A125787, A125788; A125789 (antidiagonal sums), A125714.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Dec 09 2006
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