%I #3 Mar 30 2012 18:37:02

%S 1,1,1,1,2,1,1,4,3,1,1,10,9,4,1,1,34,33,15,5,1,1,154,153,65,23,6,1,1,

%T 874,873,339,119,32,7,1,1,5914,5913,2103,719,186,42,8,1,1,46234,46233,

%U 15171,5039,1230,267,54,9,1,1,409114,409113,124755,40319,9258,1891,380

%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 (m>=1).

%C Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).

%e Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:

%e row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];

%e row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];

%e row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].

%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), ...;

%e 1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;

%e 1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;

%e 1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;

%e 1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;

%e 1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;

%e 1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;

%e 1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;

%e 1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;

%e Columns include:

%e k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);

%e k=2: A007489 (Sum of k!, k=1..n);

%e k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);

%e k=4: A033312 (n! - 1);

%e k=5: Partial sums of A001705;

%e k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).

%p {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,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

%Y Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jan 04 2007