cp's OEIS Frontend

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).

%I A127054 #3 Mar 30 2012 18:37:02
%S A127054 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 A127054 874,873,339,119,32,7,1,1,5914,5913,2103,719,186,42,8,1,1,46234,46233,
%U A127054 15171,5039,1230,267,54,9,1,1,409114,409113,124755,40319,9258,1891,380
%N 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).
%C A127054 Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).
%e A127054 Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
%e A127054 row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
%e A127054 row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
%e A127054 row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
%e A127054 The terms that are excluded in the partial sums are shown enclosed in
%e A127054 parenthesis in the table below. Rows of this table begin:
%e A127054 1,(1), 1, (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, ...;
%e A127054 1,(2), 3, (4), 5, 6, (7), 8, 9, 10, (11), 12, 13, 14, 15, (16), ...;
%e A127054 1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
%e A127054 1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
%e A127054 1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
%e A127054 1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
%e A127054 1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
%e A127054 1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
%e A127054 1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
%e A127054 1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
%e A127054 Columns include:
%e A127054 k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
%e A127054 k=2: A007489 (Sum of k!, k=1..n);
%e A127054 k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
%e A127054 k=4: A033312 (n! - 1);
%e A127054 k=5: Partial sums of A001705;
%e A127054 k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).
%p A127054 {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 A127054 Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.
%K A127054 nonn,tabl
%O A127054 0,5
%A A127054 _Paul D. Hanna_, Jan 04 2007