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Search: id:A102316
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| A102316 |
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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. |
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+0
5
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| 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
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OFFSET
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0,5
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COMMENT
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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 (callan(AT)stat.wisc.edu), Feb 04 2006
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FORMULA
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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).
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EXAMPLE
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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],...
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PROGRAM
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(PARI) T(n, k)=if(n<k|k<0, 0, if(n==0|k==0, 1, T(n, k-1)+(k+1)*T(n-1, k)))
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CROSSREFS
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Cf. A102086, A082161, A102317.
Sequence in context: A127501 A118408 A079268 this_sequence A133709 A124040 A160332
Adjacent sequences: A102313 A102314 A102315 this_sequence A102317 A102318 A102319
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jan 04 2005
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