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 A102316 Triangle, read by rows, where T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0. 4
 1, 1, 1, 1, 3, 3, 1, 7, 16, 16, 1, 15, 63, 127, 127, 1, 31, 220, 728, 1363, 1363, 1, 63, 723, 3635, 10450, 18628, 18628, 1, 127, 2296, 16836, 69086, 180854, 311250, 311250, 1, 255, 7143, 74487, 419917, 1505041, 3683791, 6173791, 6173791, 1, 511, 21940 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Main diagonal is A082161 (with offset). Row sums give A102317. T(n,k) = number of column-marked subdiagonal paths of steps east (1,0) and north (0,1) from the origin to (n,k). Subdiagonal means that the path never rises above the diagonal line y=x and column-marked means that for 1 <= i <= n, one unit square directly below the i-th east step and above the line y=-1 is marked. - David Callan, Feb 04 2006 LINKS FORMULA T(n, k) = Sum_{j=0..k} (j+1)*T(n-1, j) for n>k>0, T(n, 0)=1 for n>=0. T(n, n) = A082161(n) for n>0; A082161(n+1) = Sum_{k=0..n} (k+1)*T(n, k). EXAMPLE T(5,2) = 220 = 1*1 + 2*15 + 3*63 = 1*T(4,0) + 2*T(4,1) + 3*T(4,2). T(5,2) = 220 = 31 + 3*63 = T(5,1) + (2+1)*T(4,2). T(5,3) = 728 = 220 + 4*127 = T(5,2) + (3+1)*T(4,3). Rows begin: [1], [1,1], [1,3,3], [1,7,16,16], [1,15,63,127,127], [1,31,220,728,1363,1363], [1,63,723,3635,10450,18628,18628], [1,127,2296,16836,69086,180854,311250,311250], [1,255,7143,74487,419917,1505041,3683791,6173791,6173791],... PROG (PARI) T(n, k)=if(n

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Last modified January 25 16:42 EST 2020. Contains 331245 sequences. (Running on oeis4.)