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 A131338 Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row. 8
 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 29, 43, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 37, 51, 71, 100, 143, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 35, 46, 61, 81, 108, 145, 196 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 LINKS Paul D. Hanna, Rows n = 0..16, flattened. FORMULA T(n,k) = Sum_{i=0..k-n} T(n-1,i) for k>n, else T(n,k)=1 for n>=k>=0. Right border: T(n+1, (n+1)*(n+2)/2) = A098569(n) = Sum_{k=0..n} C( (k+1)*(k+2)/2 + n-k-1, n-k). T(n, n*(n-1)/2 + 1) = Sum_{k=0..n-1} C(k*(k+1)/2, n-k) = A121690(n-1) for n>=1. - Paul D. Hanna, Aug 30 2007 EXAMPLE Triangle begins: 1; 1, 1; 1,1, 1,2; 1,1,1, 1,2,3,5; 1,1,1,1, 1,2,3,4,6,9,14; 1,1,1,1,1, 1,2,3,4,5,7,10,14,20,29,43; 1,1,1,1,1,1, 1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,143; 1,1,1,1,1,1,1, 1,2,3,4,5,6,7,9,12,16,21,27,35,46,61,81,108,145,196,267,367,510; ... Row sums equal the row sums (A098569) of triangle A098568, where A098568(n, k) = binomial( (k+1)*(k+2)/2 + n-k-1, n-k): 1; 1, 1; 1, 3, 1; 1, 6, 6, 1; 1, 10, 21, 10, 1; 1, 15, 56, 55, 15, 1; 1, 21, 126, 220, 120, 21, 1; ... PROG (PARI) T(n, k)=if(k>n*(n+1)/2 || k<0, 0, if(k<=n, 1, sum(i=0, k-n, T(n-1, i)))) for(n=0, 10, for(k=0, n*(n+1)/2, print1(T(n, k), ", ")); print("")) CROSSREFS Cf. A098568, A098569 (row sums), A121690, A183202. Cf. A214403 (variant). Sequence in context: A220091 A063746 A201075 * A242784 A265313 A106498 Adjacent sequences:  A131335 A131336 A131337 * A131339 A131340 A131341 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jun 29 2007 STATUS approved

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