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A337841
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Triangle read by rows, T(n, k) = binomial(2*n-1, 2*k-1) * binomial(2*n-2*k, n-k) * (k+1) / binomial(n+k+1, n-k) for 0 <= k <= n.
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0
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1, 0, 2, 0, 3, 3, 0, 6, 10, 4, 0, 14, 30, 21, 5, 0, 36, 90, 84, 36, 6, 0, 99, 275, 308, 180, 55, 7, 0, 286, 858, 1092, 780, 330, 78, 8, 0, 858, 2730, 3822, 3150, 1650, 546, 105, 9, 0, 2652, 8840, 13328, 12240, 7480, 3094, 840, 136, 10, 0, 8398, 29070, 46512, 46512, 31977, 15561, 5320, 1224, 171, 11
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OFFSET
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0,3
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LINKS
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FORMULA
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T(n,k) = binomial(2*n-1,n-k) * k * (2*k+1) * (2*k+2) / ((n+k)*(n+k+1)) for 1 <= k <= n, and T(n,0) = 0^n for n >= 0.
T(n,n) = n+1 for n >= 0; T(n,n-1) = (n-1) * (2*n-1) for n > 0; T(n,n-2) = (n-1) * (n-2) * (2*n-3) for n > 1.
T(n,k) = T(n-1,k) * (2*n-2) * (2*n-1) / ((n-1) * (n+2) - (k-1) * (k+2)) for 0 <= k < n with initial values T(n,n) = n+1 for n >= 0.
Row sums are A000984(n) for n >= 0.
Alternating row sums are 0 for n > 1.
Sum_{k=0..n} (-1)^k * T(n,k) * (k*(k+1)/2)^m = 0 for 0 <= m <= n-2.
T(n,1) = 12 * binomial(2*n-1,n-1)/((n+1)*(n+2)) = A007054(n) for n > 0.
T(n,k) = T(n,1)*(k*(k+1)*(2*k+1)/6)*binomial(n-1,k-1)/binomial(n+1+k,k-1) for 1 <= k <= n.
T(n,k) = A128899(n,k) * (k+1) * (2*k+1) / (n+k+1) for 0 <= k <= n.
T(n,0) + Sum_{k=1..n} T(n,k) / (k*(k+1)) = A000108(n) for n >= 0. (End)
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EXAMPLE
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The triangle T(n,k) for 0 <= k <= n starts:
n\k: 0 1 2 3 4 5 6 7 8 9 10
=======================================================================
0 : 1
1 : 0 2
2 : 0 3 3
3 : 0 6 10 4
4 : 0 14 30 21 5
5 : 0 36 90 84 36 6
6 : 0 99 275 308 180 55 7
7 : 0 286 858 1092 780 330 78 8
8 : 0 858 2730 3822 3150 1650 546 105 9
9 : 0 2652 8840 13328 12240 7480 3094 840 136 10
10 : 0 8398 29070 46512 46512 31977 15561 5320 1224 171 11
etc.
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MAPLE
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T := proc(n, k) option remember; if k = n then n+1 else
T(n-1, k)*(2*n-2)*(2*n-1)/((n-1)*(n+2)-(k-1)*(k+2)) fi end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Nov 02 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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