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A172179
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(1,[99n+1]) Pascal Triangle.
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3
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1, 1, 100, 1, 101, 199, 1, 102, 300, 298, 1, 103, 402, 598, 397, 1, 104, 505, 1000, 995, 496, 1, 105, 609, 1505, 1995, 1491, 595, 1, 106, 714, 2114, 3500, 3486, 2086, 694, 1, 107, 820, 2828, 5614, 6986, 5572, 2780, 793, 1, 108, 927, 3648, 8442, 12600, 12558
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OFFSET
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1,3
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LINKS
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FORMULA
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T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 100, T(n,k)=0 if k<0 or if k>=n. - Philippe Deléham, Dec 26 2013
T(n, k) = 99*binomial(n-1, k-2) + binomial(n-1, k-1).
Sum_{k=1..n} T(n, k) = 100*A000225(n-1) + 1. (End)
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EXAMPLE
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Triangle begins as:
1;
1, 100;
1, 101, 199;
1, 102, 300, 298;
1, 103, 402, 598, 397;
1, 104, 505, 1000, 995, 496;
1, 105, 609, 1505, 1995, 1491, 595;
1, 106, 714, 2114, 3500, 3486, 2086, 694;
1, 107, 820, 2828, 5614, 6986, 5572, 2780, 793;
1, 108, 927, 3648, 8442, 12600, 12558, 8352, 3573, 892;
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MATHEMATICA
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Table[99*Binomial[n-1, k-2] + Binomial[n-1, k-1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Apr 27 2022 *)
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PROG
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(SageMath) flatten([[98*binomial(n-1, k-2) + binomial(n, k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 27 2022
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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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