A127054 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).
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, 874, 873, 339, 119, 32, 7, 1, 1, 5914, 5913, 2103, 719, 186, 42, 8, 1, 1, 46234, 46233, 15171, 5039, 1230, 267, 54, 9, 1, 1, 409114, 409113, 124755, 40319, 9258, 1891, 380
Offset: 0
Examples
Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, ...;
1,(2), 3, (4), 5, 6, (7), 8, 9, 10, (11), 12, 13, 14, 15, (16), ...;
1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
Columns include:
k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
k=2: A007489 (Sum of k!, k=1..n);
k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
k=4: A033312 (n! - 1);
k=5: Partial sums of A001705;
k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).
Crossrefs
Programs
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Maple
{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}
Comments