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A176123
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Irregular triangle, read by rows, T(n, k) = binomial(n-(k-1),k-1), 1 <= k <= floor(n/2-1).
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1
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1, 1, 1, 5, 1, 6, 1, 7, 15, 1, 8, 21, 1, 9, 28, 35, 1, 10, 36, 56, 1, 11, 45, 84, 70, 1, 12, 55, 120, 126, 1, 13, 66, 165, 210, 126, 1, 14, 78, 220, 330, 252, 1, 15, 91, 286, 495, 462, 210, 1, 16, 105, 364, 715, 792, 462, 1, 17, 120, 455, 1001, 1287, 924, 330, 1, 18, 136, 560, 1365, 2002, 1716, 792
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OFFSET
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4,4
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COMMENTS
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The row sum is A099572 which has limiting ratio of (1+sqrt(5))/2.
This is Pascal's triangle (A007318) read along upward sloping diagonals and truncated.
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LINKS
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EXAMPLE
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Triangle begins as:
1;
1;
1, 5;
1, 6;
1, 7, 15;
1, 8, 21;
1, 9, 28, 35;
1, 10, 36, 56;
1, 11, 45, 84, 70;
1, 12, 55, 120, 126;
1, 13, 66, 165, 210, 126;
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MAPLE
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seq(seq( binomial(n-k+1, k-1), k=1..floor((n-2)/2)), n=4..20); # G. C. Greubel, Nov 27 2019
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MATHEMATICA
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Table[Binomial[n-k+1, k-1], {n, 4, 20}, {k, Floor[(n-2)/2]}]//Flatten
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PROG
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(PARI) T(n, k) = binomial(n-k+1, k-1);
for(n=4, 20, for(k=1, (n-2)\2, print1(T(n, k), ", "))) \\ G. C. Greubel, Nov 27 2019
(Magma) [Binomial(n-k+1, k-1): k in [1..Floor((n-2)/2)], n in [4..20]]; // G. C. Greubel, Nov 27 2019
(Sage) [[binomial(n-k+1, k-1) for k in (1..floor((n-2)/2))] for n in (4..20)] # G. C. Greubel, Nov 27 2019
(GAP) Flat(List([4..20], n-> List([1..Int((n-2)/2)], k-> Binomial(n-k+1, k-1) ))); # G. C. Greubel, Nov 27 2019
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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