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A117207
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Number triangle read by rows: T(n,k)=sum{j=0..n-k, C(n+j,j+k)C(n-j,k)}.
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1
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1, 3, 1, 10, 7, 1, 35, 31, 13, 1, 126, 121, 81, 21, 1, 462, 456, 381, 181, 31, 1, 1716, 1709, 1583, 1058, 358, 43, 1, 6435, 6427, 6231, 5055, 2605, 645, 57, 1, 24310, 24301, 24013, 21661, 14605, 5785, 1081, 73, 1, 92378, 92368, 91963, 87643, 70003, 38251, 11791
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OFFSET
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0,2
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COMMENTS
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The number of different ordered partitions of n+1 into n+1 bins (as with A001700), such that more than k bins are nonempty. - Dan Uznanski, Jan 23 2012
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LINKS
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FORMULA
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T(n,k)=C(2n+1,n+1)-sum{j=1..k, product{i=0..j-2, (n-i)^2}/((j-1)!j!)}}*(n+1).
T(n,k)=[x^(n-k)](1+x)^(n-k)*F(-n-1,-n,1,x/(1+x)). - Paul Barry, Oct 01 2010
T(n,k)=C(2n+1,n+1)-(n+1)*sum(j=1,k, C(n,j-1)^2/j). - M. F. Hasler, Jan 25 2012
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EXAMPLE
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Triangle begins
1,
3, 1,
10, 7, 1,
35, 31, 13, 1,
126, 121, 81, 21, 1,
462, 456, 381, 181, 31, 1,
1716, 1709, 1583, 1058, 358, 43, 1
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MATHEMATICA
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Table[Sum[Binomial[n+j, j+k]Binomial[n-j, k], {j, 0, n-k}], {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Apr 23 2016 *)
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PROG
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(PARI) T(n, k)=sum(j=0, n-k, binomial(n+j, j+k)*binomial(n-j, k))
T(n, k)=binomial(2*n+1, n+1)-(n+1)*sum(j=1, k, binomial(n, j-1)^2/j)
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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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