A156628 - OEIS

%I #5 Jun 23 2021 02:21:28

%S 1,1,1,3,2,1,13,7,3,1,71,33,13,4,1,461,191,71,20,5,1,3447,1297,461,

%T 120,28,6,1,29093,10063,3447,836,181,38,7,1,273343,87669,29093,6616,

%U 1333,270,49,8,1,2829325,847015,273343,58576,11029,2150,375,61,9,1

%N Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

%F Column 0 = Column 0 of triangle A104980 = A003319.

%F Column 1 = Column 1 of triangle A104980 = A104981.

%F Column 3 = column 1 of A104988 (matrix square of A104980).

%F Column 5 = column 2 of A104988 (matrix square of A104980).

%e To generate the array, start with all 1's in row 0; from then on,

%e obtain row n+1 from row n by first removing terms in row n at

%e positions 0 and {(m+1)*(m+2)/2-2,m>0} and then taking partial sums.

%e This square array A begins:

%e (1), (1), 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), (7), 13, 20, (28), 38, 49, 61, (74), 89, 105, 122, 140, (159),...;

%e (13), (33), 71, 120, (181), 270, 375, 497, (637), 817, 1019, 1244, ...;

%e (71), (191), 461, 836, (1333), 2150, 3169, 4413, (5906), 8001, ...;

%e (461), (1297), 3447, 6616, (11029), 19030, 29483, 42775, (59324),...;

%e (3447), (10063), 29093, 58576, (101351), 185674, 300329, 451277, ...;

%e (29093), (87669), 273343, 573672, (1024949), 1982310, 3330651, ...;

%e (273343), (847015), 2829325, 6159976, (11320359), 23009602, 39998897, ...;

%e where terms in parenthesis at positions {0,1,4,8,13,..} in a row

%e are removed before taking partial sums to obtain the next row.

%e ...

%e RELATION TO SPECIAL TRIANGLE.

%e Triangle A104980 begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 13, 7, 3, 1;

%e 71, 33, 13, 4, 1;

%e 461, 191, 71, 21, 5, 1;

%e 3447, 1297, 461, 133, 31, 6, 1;

%e 29093, 10063, 3447, 977, 225, 43, 7, 1; ...

%e in which column 0 and column 1 are found in square array A.

%e ...

%e Matrix square of A104980 = triangle A104988 which begins:

%e 1;

%e 2, 1;

%e 8, 4, 1;

%e 42, 20, 6, 1;

%e 266, 120, 38, 8, 1;

%e 1954, 836, 270, 62, 10, 1;

%e 16270, 6616, 2150, 516, 92, 12, 1;

%e 151218, 58576, 19030, 4688, 882, 128, 14, 1; ...

%e where column 1 and column 2 are also found in square array A.

%o (PARI) {T (n, k)=local (A=0, b=2, c=1, 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. columns: A003319, A104981, A156629, related triangles: A104980, A104988.

%Y Cf. related tables: A136212, A136213, A125714, A135876, A127054, A125781, A136217.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Feb 17 2009